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Author Description

Mark L. Lewis is an American mathematician who has made significant contributions to group theory, particularly in the study of characters of finite groups. He has served as a professor in the Department of Mathematical Sciences at Kent State University, where he has published extensively on representation theory and related areas of algebra.

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Character degree graphs that are complete graphs

2006-08-31
Mariagrazia Bianchi , David Chillag , Mark L. Lewis +1 more
Let $G$ be a finite group, and write $\operatorname {cd}(G)$ for the set of degrees of irreducible characters of $G$. We define $\Gamma (G)$ to be the graph whose vertex … Let $G$ be a finite group, and write $\operatorname {cd}(G)$ for the set of degrees of irreducible characters of $G$. We define $\Gamma (G)$ to be the graph whose vertex set is $\operatorname {cd}(G)-\{1\}$, and there is an edge between $a$ and $b$ if $(a,b)>1$. We prove that if $\Gamma (G)$ is a complete graph, then $G$ is a solvable group.
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An Overview of Graphs Associated with Character Degrees And Conjugacy Class Sizes in Finite Groups

2008-02-01
Mark L. Lewis
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Solvable groups whose degree graphs have two connected components

2001-01-05
Mark L. Lewis

Connectedness of degree graphs of nonsolvable groups

2003-06-30
Mark L. Lewis , D. L. White

Character Theory of Finite Groups

2010-01-01
Mark L. Lewis , Gabriel Navarro , D. S. Passman +1 more
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Codegrees and nilpotence class of <i>p</i>-groups

2015-12-17
Ni Du , Mark L. Lewis
Abstract If χ is an irreducible character of a finite group G , then the codegree of χ is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> … Abstract If χ is an irreducible character of a finite group G , then the codegree of χ is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>|</m:mo> <m:mo>/</m:mo> <m:mi>χ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${|G:{\rm ker}(\chi)|/\chi(1)}$ . We show that if G is a p -group, then the nilpotence class of G is bounded in terms of the largest codegree for an irreducible character of G .

Nonsolvable groups with no prime dividing three character degrees

2011-05-09
Mark L. Lewis , D. L. White
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Isomorphism in expanding families of indistinguishable groups

2012-01-01
Mark L. Lewis , James B. Wilson
For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients … For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ pairwise nonisomorphic quotients of order $p^{n}$. Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most $p$. They are also directly and centrally indecomposable and of the same indecomposability type. The recognized portions of their automorphism groups are isomorphic, represented isomorphically on their abelianizations, and of small index in their full automorphism groups. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.
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Determining Group Structure from Sets of Irreducible Character Degrees

1998-08-01
Mark L. Lewis

A solvable group whose character degree graph has diameter $3$

2001-06-20
Mark L. Lewis
We show that there is a solvable group $G$ so that the character degree graph of $G$ has diameter $3$. We show that there is a solvable group $G$ so that the character degree graph of $G$ has diameter $3$.
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The vanishing-off subgroup

2008-12-19
Mark L. Lewis
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SOLVABLE GROUPS WITH CHARACTER DEGREE GRAPHS HAVING 5 VERTICES AND DIAMETER 3

2002-12-11
Mark L. Lewis
ABSTRACT Let G be a finite group and cd(G) the character degrees of G. The degree graph Δ(G) of G is the graph whose vertices are the primes dividing degrees … ABSTRACT Let G be a finite group and cd(G) the character degrees of G. The degree graph Δ(G) of G is the graph whose vertices are the primes dividing degrees in cd(G), and there is an edge between p and q if pq divides some degree in cd(G). In this paper, we show that if Δ(G) has 5 vertices, then the diameter of Δ(G) is at most 2. This shows that the example in[9] of a solvable group G where Δ(G) has diameter 3 has the fewest number of vertices possible.

A character theoretic condition characterizing nilpotent groups

1999-01-01
Stephen M. Gagola , Mark L. Lewis

A characterization of the prime graphs of solvable groups

2014-09-17
A. Gruber , Thomas Michael Keller , Mark L. Lewis +2 more
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Non-divisibility among character degrees

2005-01-19
Mark L. Lewis , Alexander Moretó , Thomas R. Wolf
In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the … In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the condition no divisibility among degrees (NDAD). We conjecture that the number of character degrees of a group that satisfies NDAD is bounded and we prove this for solvable groups. More precisely, we prove that solvable groups with NDAD have at most four character degrees and have derived length at most 3. We give a group-theoretic characterization of the solvable groups satisfying NDAD with four character degrees. Since the structure of groups with at most three character degrees is known, these results describe the structure of solvable groups with NDAD.

Diameters of degree graphs of nonsolvable groups

2004-10-05
Mark L. Lewis , D. L. White

Bounding Fitting Heights of Character Degree Graphs

2001-08-01
Mark L. Lewis

Generalizing Camina groups and their character tables

2008-11-07
Mark L. Lewis
We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are … We generalize the definition of Camina groups. We show that our generalized Camina groups are exactly the groups isoclinic to Camina groups, and so many properties of Camina groups are shared by these generalized Camina groups. In particular, we show that if G is a nilpotent, generalized Camina group then its nilpotence class is at most 3. We use the information we collect about generalized Camina groups with nilpotence class 3 to characterize the character tables of these groups.
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p-Parts of character degrees and the index of the Fitting subgroup

2014-05-13
Mark L. Lewis , Gabriel Navarro , Thomas R. Wolf
In a solvable group G, if p2 does not divide χ(1) for all χ∈Irr(G), then we prove that |G:F(G)|p≤p2. This bound is best possible. In a solvable group G, if p2 does not divide χ(1) for all χ∈Irr(G), then we prove that |G:F(G)|p≤p2. This bound is best possible.

Classifying character degree graphs with 5 vertices

2004-11-24
Mark L. Lewis
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Groups with exactly two supercharacter theories

2016-10-07
Shawn T. Burkett , Jonathan Lamar , Mark L. Lewis +1 more
In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are ℤ3 and S3. We also show … In this paper, we classify those finite groups with exactly two supercharacter theories. We show that the solvable groups with two supercharacter theories are ℤ3 and S3. We also show that the only nonsolvable group with two supercharacter theories is Sp(6,2).
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Square character degree graphs yield direct products

2011-10-18
Mark L. Lewis , Qingyun Meng
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<i>p</i>-parts of character degrees

2015-08-31
Mark L. Lewis , Gabriel Navarro , Pham Huu Tiep +1 more
We show that if p is an odd prime and G is a finite group satisfying the condition that p 2 divides the degree of no irreducible character of G, … We show that if p is an odd prime and G is a finite group satisfying the condition that p 2 divides the degree of no irreducible character of G, then | G : O p ( G ) | p ⩽ p 4 , where O p ( G ) is the largest normal p-subgroup of G, and if P is a Sylow p-subgroup of G, then P ' ' is subnormal in G. Our investigations suggest that if p a is the largest power of p dividing the degrees of irreducible characters of G, then | G : O p ( G ) | p is bounded by p f ( a ) , where f ( a ) is a function in a and P ( a + 1 ) is subnormal in G.

Solvable Groups Having Almost Relatively Prime Distinct Irreducible Character Degrees

1995-05-01
Mark L. Lewis

On p-group Camina pairs

2012-01-01
Mark L. Lewis
Abstract. Let Abstract. Let
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Four-vertex degree graphs of nonsolvable groups

2013-01-02
Mark L. Lewis , D. L. White

Classifying Camina groups: A theorem of Dark and Scoppola

2014-04-01
Mark L. Lewis
Recall that a group $G$ is a Camina group if every nonlinear irreducible character of $G$ vanishes on $G \setminus G'$. Dark and Scoppola classified the Camina groups that can … Recall that a group $G$ is a Camina group if every nonlinear irreducible character of $G$ vanishes on $G \setminus G'$. Dark and Scoppola classified the Camina groups that can occur. We present a different proof of this classification using Theorem 2, which strengthens a result of Isaacs on Camina pairs. Theorem 2 is of independent interest.
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Finite groups with an irreducible character of large degree

2015-10-01
Hung Ngoc Nguyen , Mark L. Lewis , A. A. Schaeffer Fry
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Blocks of small defect in alternating groups and squares of Brauer character degrees

2017-07-18
Xiaoyou Chen , James P. Cossey , Mark L. Lewis +1 more
Abstract Let p be a prime. We show that other than a few exceptions, alternating groups will have p -blocks with small defect for p equal to 2 or 3. … Abstract Let p be a prime. We show that other than a few exceptions, alternating groups will have p -blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p -subgroup P and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>P</m:mi> </m:mrow> </m:math> {G/P} is nilpotent if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>φ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> {\varphi(1)^{2}} divides <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>φ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:math> {|G:{\rm ker}(\varphi)|} for every irreducible Brauer character φ of G .

Fitting heights and the character degree graph

2000-11-01
Mark L. Lewis

Characters of Maximal Subgroups ofM-Groups

1996-08-01
Mark L. Lewis

The cyclic graph (deleted enhanced power graph) of a direct product

2021-03-04
David G. Costanzo , Mark L. Lewis , Stefano Schmidt +2 more
Let $G$ be a finite group. Define a graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to be adjacent if and … Let $G$ be a finite group. Define a graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to be adjacent if and only if $\langle x,y\rangle$ is cyclic. Denote this graph by $\Delta(G)$. The graph $\Delta(G)$ has appeared in the literature under the names cyclic graph and deleted enhanced power graph. If $G$ and $H$ are nontrivial groups, then $\Delta(G\times H)$ is completely characterized. In particular, if $\Delta(G\times H)$ is connected, then a diameter bound is obtained, along with an example meeting this bound. Also, necessary and sufficient conditions for the disconnectedness of $\Delta(G\times H)$ are established.
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Derived lengths and character degrees

1998-01-01
Mark L. Lewis
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite solvable group. Assume that the degree graph of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite solvable group. Assume that the degree graph of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has exactly two connected components that do not contain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Suppose that one of these connected components contains the subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet a 1 comma ellipsis comma a Subscript n Baseline EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{ a_{1}, \dots , a_{n} \}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript i"> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">a_{i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript j"> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">a_{j}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are coprime when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i not-equals j"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i \not = j</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then the derived length of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is less than or equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue c d left-parenthesis upper G right-parenthesis EndAbsoluteValue minus n plus 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>cd</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">|\operatorname {cd}(G)|-n+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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Character degree graphs of automorphism groups of characteristically simple groups

2008-11-07
Mark L. Lewis , John K. McVey
Throughout this note, G will be a finite group, IrrðGÞ will be the set of irreducible characters of G, cdðGÞ will be the set of character degrees of G, and … Throughout this note, G will be a finite group, IrrðGÞ will be the set of irreducible characters of G, cdðGÞ will be the set of character degrees of G, and rðGÞ will be the set of primes that divide degrees in cdðGÞ.The prime vertex degree graph of G, written DðGÞ, is the graph with rðGÞ as its vertex set, and with an edge between p and q if pq divides some degree a A cdðGÞ.An overview of the literature on these graphs can be found in [2].It seems that most groups G have graphs DðGÞ that are complete graphs (although we do not want to be precise on what we mean by 'most').For solvable groups, it has been shown that if DðGÞ is not complete, then the structure of G is limited (see [4] and[6]).In this note, we show that a large class of non-solvable groups have degree graphs that are complete.To study the degree graphs of non-solvable groups, a standard starting point is with the degree graphs of non-abelian simple groups.In a series of papers, Don White has classified the character degree graphs of all non-abelian simple groups.Except for some examples of small rank, nearly all non-abelian simple groups have degree graphs that are complete graphs.These results are summarized in [5].Extending this idea, this note looks at the characteristically simple groups, groups for which there are exactly two characteristic subgroups.Each such group is a direct product of copies of a fixed simple group.With this in mind, we let G n denote G Â Á Á Á Â G (n copies); thus S n is characteristically simple when S itself is a simple group.We now fix S to be a non-abelian simple group.Because the non-abelian simple groups have already been addressed, we restrict our attention to when n > 1.However, it is relatively easy to show that DðS n Þ is complete for n > 1.More generally, the graph DðH n Þ associated with any non-abelian group H is always complete for every integer n > 1.Thus we expand our scope and consider extending these groups.We recognize first that S n is isomorphic to InnðS n Þ. Identifying S n with its set of inner automorphisms, our goal herein is to prove the following result.
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The Number of Irreducible Character Degrees of Solvable Groups Satisfying the One-Prime Hypothesis

2005-10-01
Mark L. Lewis

Nonsolvable Groups Satisfying the One-prime Hypothesis

2007-04-17
Mark L. Lewis , D. L. White

DERIVED LENGTHS OF SOLVABLE GROUPS SATISFYING THE ONE-PRIME HYPOTHESIS. III

2001-01-01
Mark L. Lewis
A finite group G is said to satisfy the one-prime hypothesis if the greatest common divisor of any two distinct irreducible character degrees is either 1 or a prime number. … A finite group G is said to satisfy the one-prime hypothesis if the greatest common divisor of any two distinct irreducible character degrees is either 1 or a prime number. The principal result of this paper is that if G is solvable and satisfies the one-prime hypothesis, and if G has a nonabelian nilpotent homomorphic image, then the derived length of G is at most 4.

Classifying character degree graphs with six vertices

2019-02-06
Mark W. Bissler , Jacob Laubacher , Mark L. Lewis

THE CYCLIC GRAPH OF A <i>Z</i>-GROUP

2020-12-14
David G. Costanzo , Mark L. Lewis , Stefano Schmidt +2 more
For a group $G$, we define a graph $\Delta(G)$ by letting $G^{\#} = G \setminus \{ 1 \}$ be the set of vertices and by drawing an edge between distinct … For a group $G$, we define a graph $\Delta(G)$ by letting $G^{\#} = G \setminus \{ 1 \}$ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\#}$ if and only if the subgroup $\langle x,y\rangle$ is cyclic. Recall that a $Z$-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta(G)$ for a $Z$-group $G$.
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Transitive permutation groups in which all derangements are involutions

2005-12-22
I. M. Isaacs , Thomas Michael Keller , Mark L. Lewis +1 more
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Generalizing Baer’s norm

2018-08-08
Mark L. Lewis , Mohammad Zarrin
Abstract Let G be a group and n a positive integer. We define <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {B_{n}(G)} to … Abstract Let G be a group and n a positive integer. We define <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {B_{n}(G)} to be the intersection of the normalizers of all the non- n -subnormal subgroups of G . We give a new characterization for nilpotent groups in terms of a series defined via <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>B</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {B_{n}(G)} .

Solvable groups whose prime divisor character degree graphs are 1-connected

2019-03-01
Mark L. Lewis , Qingyun Meng

Groups where the centers of the irreducible characters form a chain

2020-01-01
Mark L. Lewis
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Nonsolvable Groups All of Whose Character Degrees are Odd-Square-Free

2011-03-21
Mark L. Lewis , D. L. White
Abstract A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G … Abstract A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S ≤ G ≤ Aut(S) for a finite simple group S. More generally, we show that if G is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups. If the alternating group A 7 is involved in G, the structure of G can be further restricted. Key Words: Character degreesNonsolvable groupsSimple groupsSquare-free2000 Mathematics Subject Classification: 20C15 Notes α = ε + ε2 + ε4, ε =e 2πi/7. Communicated by P. Tiep.
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Bounding group orders by large character degrees: A question of Snyder

2014-02-10
Mark L. Lewis
Abstract Let G be a nonabelian finite group and let d be an irreducible character degree of G . Then there is a positive integer e so that | G … Abstract Let G be a nonabelian finite group and let d be an irreducible character degree of G . Then there is a positive integer e so that | G | = d ( d + e ). Snyder has shown that if e &gt; 1, then | G | is bounded by a function of e . This bound has been improved by Isaacs and by Durfee and Jensen. In this paper we will show, for groups having a nontrivial, abelian normal subgroup, that | G | ≤ e 4 - e 3 . Given that there are a number of solvable groups that meet this bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.
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Groups with exactly one irreducible character of degree divisible by<i>p</i>

2014-05-18
Daniel Goldstein , Robert M. Guralnick , Mark L. Lewis +3 more
Let p be a prime.We characterize those finite groups which have precisely one irreducible character of degree divisible by p.Minimal situations constitute a classical theme in group theory.Not only do … Let p be a prime.We characterize those finite groups which have precisely one irreducible character of degree divisible by p.Minimal situations constitute a classical theme in group theory.Not only do they arise naturally, but they also provide valuable hints in searching for general patterns.In this paper, we are concerned with character degrees.One of the key results on character degrees is the Itô-Michler theorem, which asserts that a prime p does not divide the degree of any complex irreducible character of a finite group G if and only if G has a normal, abelian Sylow p-subgroup.In [Isaacs et al. 2009], Isaacs together with the fourth, fifth, and sixth authors of this paper studied the finite groups that have only one character degree divisible by p.They proved, among other things, that the Sylow p-subgroups of those groups were metabelian.This suggested that the derived length of the Sylow p-subgroups might be related with the number of different character degrees divisible by p.However, nothing could be said in [Isaacs et al. 2009] on how large p-Sylow normalizers were inside G. (As a trivial example, the dihedral group of order 2n for n odd has a unique character degree divisible by 2, and a self-normalizing Sylow 2-subgroup of order 2.)In this paper, we go further and completely classify the finite groups with exactly one irreducible character of degree divisible by p.Our focus now therefore is not only on the set of character degrees but also on the multiplicity of the number of irreducible characters of each degree.In Section 1, we define the terms semiextraspecial, ultraspecial, and doubly transitive Frobenius groups of Dickson type.
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CAMINA p-groups that are generalized Frobenius complements

2015-04-07
I. M. Isaacs , Mark L. Lewis
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OBTAINING NUCLEI FROM CHAINS OF NORMAL SUBGROUPS

2006-04-01
Mark L. Lewis
In this paper, we reexamine the foundation of Isaacs' π-theory. One of the key concepts in Isaacs' π-theory is the construction of the characters B π (G) for a π-separable … In this paper, we reexamine the foundation of Isaacs' π-theory. One of the key concepts in Isaacs' π-theory is the construction of the characters B π (G) for a π-separable group G. The key to determining which characters lie in B π (G) was the construction of a nucleus for each irreducible character χ. In this paper, we present a different way of finding a nucleus for χ which is based on a chain of normal subgroup [Formula: see text]. Using this nucleus, we obtain the set of characters [Formula: see text]. We investigate the properties that [Formula: see text] has in common with B π (G).

Nonabelian Fully-Ramified Sections

1996-10-01
Mark L. Lewis
Abstract Let G be a finite group and let K and L be normal subgroups of G such that | K : L | and | G : K | … Abstract Let G be a finite group and let K and L be normal subgroups of G such that | K : L | and | G : K | are relatively prime, and assume that | K : L | is odd. Let H be a subgroup of G such that G = HK and H ∩ K = L . Let φ be an irreducible character of L that is invariant under the action of L and is fully ramified with respect to K/L . If χ ∈ Irr( G ) is a constituent of φ G , then we prove that χ H has a unique irreducible constituent having odd multiplicity.
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Supercharacter theories of semiextraspecial p-groups and Frobenius groups

2018-02-13
Mark L. Lewis , Casey Wynn

$M_{p}$ -GROUPS AND BRAUER CHARACTER DEGREES

2025-05-09
Xiaoyou Chen , Mark L. Lewis
Abstract Let G be a finite group and p be a prime. We prove that if G has three codegrees, then G is an M -group. We prove for some … Abstract Let G be a finite group and p be a prime. We prove that if G has three codegrees, then G is an M -group. We prove for some prime p that if the degree of every nonlinear irreducible Brauer character of G is a prime, then for every normal subgroup N of G , either $G/N$ or N is an $M_p$ -group.

On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups

2025-03-29
Yiftach Barnea , Rachel D. Camina , Mikhail Ershov +1 more

On Common Zeros of Characters of Finite Groups

2025-02-20
Mark L. Lewis , Lucia Morotti , Emanuele Pacifici +2 more

Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$

2025-02-05
Mark L. Lewis , Shannon M. Tefft
Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call … Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call $D$ the Ducci function and the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$ for $\mathbf{u} \in \mathbb{Z}_m^n$. Every Ducci sequence enters a cycle, so we can let $\text{Per}(\mathbf{u})$ be the number of tuples in the Ducci cycle of $\mathbf{u}$, or the period of $\mathbf{u}$. In this paper, we will look at what different possible values of $\text{Per}(\mathbf{u})$ we can have and some conditions that if $\mathbf{u}$ meets at least one of them, $\mathbf{u}$ will generate a period smaller than the maximum period.
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Examining $H$-Closed Ducci Sequences on $\mathbb{Z}_m^n$

2025-02-05
Mark L. Lewis , Shannon M. Tefft
Let $D$ be an endomorphism on $\mathbb{Z}_m^n$ so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call … Let $D$ be an endomorphism on $\mathbb{Z}_m^n$ so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u} \in \mathbb{Z}_m^n$, which always enters a cycle. Now let $H$ be an endomorphism on $\mathbb{Z}_m^n$ such that \[H(x_1, x_2, ..., x_n)=(x_2, x_3, ..., x_n, x_1).\] In this paper, we will talk about a few cases when $\mathbf{u}$ and $H^{\beta}(\mathbf{u})$ have the same Ducci cycle for $\beta > 0$, as well as prove a few cases of $n,m$ where this is guaranteed for every $\mathbf{u} \in \mathbb{Z}_m^n$.
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Neighborhoods, connectivity, and diameter of the nilpotent graph of a finite group

2025-02-05
Costantino Delizia , Michele Gaeta , Mark L. Lewis +1 more
The nilpotent graph of a group $G$ is the simple and undirected graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if they generate a … The nilpotent graph of a group $G$ is the simple and undirected graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if they generate a nilpotent subgroup of $G$. Here we discuss some topological properties of the nilpotent graph of a finite group $G$. Indeed, we characterize finite solvable groups whose closed neighborhoods are nilpotent subgroups. Moreover, we study the connectivity of the graph $\Gamma(G)$ obtained removing all universal vertices from the nilpotent graph of $G$. Some upper bounds to the diameter of $\Gamma(G)$ are provided when $G$ belongs to some classes of groups.
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Semi-extraspecial $p$-groups with automorphisms of large order

2025-02-03
Sofia Brenner , Rachel D. Camina , Mark L. Lewis
In this paper, we consider semi-extraspecial $p$-groups $G$ that have an automorphism of order $|G:G'| - 1$. We prove that these groups are isomorphic to Sylow $p$-subgroups of ${\rm SU}_3 … In this paper, we consider semi-extraspecial $p$-groups $G$ that have an automorphism of order $|G:G'| - 1$. We prove that these groups are isomorphic to Sylow $p$-subgroups of ${\rm SU}_3 (p^{2a})$ for some integer $a$. If $p$ is odd, this is equivalent to saying that $G$ is isomorphic to a Sylow $p$-subgroup of ${\rm SL}_3 (p^a)$.
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ON NEIGHBOURHOODS IN THE ENHANCED POWER GRAPH ASSOCIATED WITH A FINITE GROUP

2024-12-02
Mark L. Lewis , Carmine Monetta
Abstract We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p -groups with the … Abstract We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p -groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.
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On non self-normalizing subgroups

2024-11-27
Mariagrazia Bianchi , Rachel D. Camina , Mark L. Lewis +2 more
Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. … Let $n$ be a non negative integer, and define $D_n$ to be the family of all finite groups having precisely $n$ conjugacy classes of nontrivial subgroups that are not self-normalizing. We are interested in studying the behavior of $D_n$ and its interplay with solvability and nilpotency. We first show that if $G$ belongs to $D_n$ with $n \le 3$, then $G$ is solvable of derived length at most 2. We also show that $A_5$ is the unique nonsolvable group in $D_4$, and that $SL_2(3)$ is the unique solvable group in $D_4$ whose derived length is larger than 2. For a group $G$, we define $D(G)$ to be the number of conjugacy classes of nontrivial subgroups that are not self-normalizing. We determine the relationship between $D(H \times K)$ and $D(H)$ and $D(K)$. We show that if $G$ is nilpotent and lies in $D_n$, then $G$ has nilpotency class at most $n/2$ and its derived length is at most $\log_2 (n/2) + 1$. We consider $D_n$ for several classes of Frobenius groups, and we use this classification to classify the groups in $D_0$, $D_1$, $D_2$, and $D_3$. Finally, we show that if $G$ is solvable and lies in $D_n$ with $n \ge 3$, then $G$ has derived length at most the minimum of $n-1$ and $3 \log_2 (n+1) + 9$.
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On the group pseudo-algebra of finite groups

2024-11-25
Mark L. Lewis , Quanfu Yan
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Groups with a Fixed Character Degree

2024-11-13
Mark L. Lewis , Brandon Martin
Let $G$ be a finite group, and let $d$ be the degree of an irreducible character of $G$ such that $|G|=d(d+e)$ for some $e>1$. Consider the case when $G$ is … Let $G$ be a finite group, and let $d$ be the degree of an irreducible character of $G$ such that $|G|=d(d+e)$ for some $e>1$. Consider the case when $G$ is solvable, $d$ is square-free, and $(d,d+e)=1$. We wish to explore an equivalent condition on $G$ when $d\in\text{cd}(G)$. We show that if $d\in\text{cd}(G)$ then there is a sequence of congruences relating the prime power factors of $d+e$ to the product of prime factors of $d$ such that the product of the moduli in this sequence of congruences is $d$. Moreover, the argument will hold in both directions.
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Some results on variations on the norm of finite groups

2024-11-12
Mark L. Lewis , Zhencai Shen , Quanfu Yan
Abstract Let G be a finite group and $$N_{\Omega }(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>Ω</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the intersection of the normalizers … Abstract Let G be a finite group and $$N_{\Omega }(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>Ω</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> be the intersection of the normalizers of all subgroups belonging to the set $$\Omega (G),$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where $$\Omega (G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is a set of all subgroups of G which have some theoretical group property. In this paper, we show that $$N_{\Omega }(G)= Z_{\infty }(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>Ω</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> if $$\Omega (G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is one of the following: (i) the set of all self-normalizing subgroups of G ; (ii) the set of all subgroups of G satisfying the subnormalizer condition in G ; (iii) the set of all pronormal subgroups of G ; (iv) the set of all weakly normal subgroups of G ; (v) the set of all NE -subgroups of G .
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The Maximum Length for Ducci Sequences on $\mathbb{}Z_m^n$ when $n$ is Even

2024-10-23
Mark L. Lewis , Shannon M. Tefft
Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] $D$ is known … Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] $D$ is known as the Ducci function and for $\mathbf{u} \in \mathbb{Z}_m^n$, $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ is the Ducci sequence of $\mathbf{u}$. Every Ducci sequence enters a cycle because $\mathbb{Z}_m^n$ is finite. In this paper, we aim to establish an upper bound for how long it will take for a Ducci sequence in $\mathbb{Z}_m^n$ to enter its cycle when $n$ is even.
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On the sum of character codegrees of finite groups

2024-10-21
Mark L. Lewis , Quanfu Yan

FINITE GROUPS WHOSE CHARACTER CODEGREES ARE CONSECUTIVE INTEGERS

2024-10-14
Mark L. Lewis , Quanfu Yan
Abstract Let G be a finite group. We investigate the structure of finite groups whose irreducible character codegrees are consecutive integers. Abstract Let G be a finite group. We investigate the structure of finite groups whose irreducible character codegrees are consecutive integers.
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On common zeros of characters of finite groups

2024-09-23
Mark L. Lewis , Lucia Morotti , Emanuele Pacifici +2 more
Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ … Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $\chi\in\text{Irr}(G)$ such that $\chi(g)=0$ (i.e., $g$ is a zero of $\chi$) and, in this case, the conjugacy class $g^G$ of $g$ in $G$ is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group $G$ such that every non-linear $\chi\in\text{Irr}(G)$ vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.
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A family of graphs that cannot occur as character degree graphs of solvable groups

2024-09-14
Mark W. Bissler , Jacob Laubacher , Mark L. Lewis

On neighborhoods in the enhanced power graph associated with a finite group

2024-08-29
Mark L. Lewis , Carmine Monetta
This article investigates neighborhoods' sizes in the enhanced power graph (as known as the cyclic graph) associated with a finite group. In particular, we characterize finite $p$-groups with the smallest … This article investigates neighborhoods' sizes in the enhanced power graph (as known as the cyclic graph) associated with a finite group. In particular, we characterize finite $p$-groups with the smallest maximum size for neighborhoods of nontrivial element in its enhanced power graph.
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A Note on the Codegree of Finite Groups

2024-08-02
Mark L. Lewis , Quanfu Yan
Abstract Let $$\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>χ</mml:mi></mml:math> be an irreducible character of a group G , and $$S_c(G)=\sum _{\chi \in \textrm{Irr}(G)}\textrm{cod}(\chi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mi>χ</mml:mi><mml:mo>∈</mml:mo><mml:mtext>Irr</mml:mtext><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mtext>cod</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>χ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> be the sum of the codegrees of … Abstract Let $$\chi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>χ</mml:mi></mml:math> be an irreducible character of a group G , and $$S_c(G)=\sum _{\chi \in \textrm{Irr}(G)}\textrm{cod}(\chi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∑</mml:mo><mml:mrow><mml:mi>χ</mml:mi><mml:mo>∈</mml:mo><mml:mtext>Irr</mml:mtext><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:msub><mml:mtext>cod</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>χ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> be the sum of the codegrees of the irreducible characters of G . Write $$\textrm{fcod} (G)=\frac{S_c(G)}{|G|}.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>fcod</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>G</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mfrac><mml:mo>.</mml:mo></mml:mrow></mml:math> We aim to explore the structure of finite groups in terms of $$\textrm{fcod} (G).$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>fcod</mml:mtext><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math> On the other hand, we determine the lower bound of $$S_c(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> for nonsolvable groups and prove that if G is nonsolvable, then $$S_c(G)\geqslant S_c(A_5)=68,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>G</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>⩾</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math> with equality if and only if $$G\cong A_5.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>≅</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math> Additionally, we show that there is a solvable group so that it has the codegree sum as $$A_5.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:mo>.</mml:mo></mml:mrow></mml:math>

Finite solvable tidy groups whose orders are divisible by two primes

2024-06-24
Nicolas F. Beike , Rachel Carleton , David G. Costanzo +4 more
Abstract In this paper, we investigate finite solvable tidy groups. We classify the tidy $$\{ p, q \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> -groups. Combining … Abstract In this paper, we investigate finite solvable tidy groups. We classify the tidy $$\{ p, q \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> -groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.
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Characterizing finite groups whose enhanced power graphs have universal vertices

2024-04-29
David G. Costanzo , Mark L. Lewis , Stefano Schmidt +2 more
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Corrigendum to “A Frobenius group analog for Camina triples” [J. Algebra 568 (2021) 160–180]

2024-04-04
Shawn T. Burkett , Mark L. Lewis

Ducci on $\mathbb{Z}_m^n$ and the Maximum Length for $n$ Odd

2024-03-08
Mark L. Lewis , Shannon M. Tefft
Define the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ so $D(x_1,x_2, ...,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$. In this paper, we will … Define the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ so $D(x_1,x_2, ...,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$. In this paper, we will prove that for $n$ odd and $m=2^lm_1$ where $m_1$ is odd, then the largest the length of a tuple can be is $l$ and $K(\mathbb{Z}_m^n)=\{(x_1, x_2, ..., x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.
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Some results on the norm of finite groups

2024-02-20
Mark L. Lewis , Zhencai Shen , Quanfu Yan
Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups … Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this paper, we show that $N_{\Omega}(G)= Z_{\infty}(G)$ if $\Omega(G)$ is one of the following: (i) the set of all self-normalizing subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of $G$; (iv) the set of all $\mathscr{H}$-subgroups of $G$; (v) the set of all weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$.
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A lower bound on the size of maximal abelian subgroups

2024-02-19
Mark L. Lewis
Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so that $|G:Z(G)| = p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let … Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so that $|G:Z(G)| = p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let $$p^l = {\rm max} \{|Z(C_G (g)):Z(G)| : g \in G \setminus Z(G)\},$$ $$p^b = {\rm max} \{|cl(g)| : g \in G \setminus Z(G) \},$$ and $p^a = |A:Z(G)|$. Then we show that $a \ge n/(b+l)$.
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On the group pseudo-algebra of finite groups

2024-02-19
Mark L. Lewis , Quanfu Yan
Let $G$ be a finite group. The group pseudo-algebra of $G$ is defined as the multi-set $C(G)=\{(d,m_G(d))\mid d\in{\rm Cod}(G)\},$ where $m_G(d)$ is the number of irreducible characters of with codegree … Let $G$ be a finite group. The group pseudo-algebra of $G$ is defined as the multi-set $C(G)=\{(d,m_G(d))\mid d\in{\rm Cod}(G)\},$ where $m_G(d)$ is the number of irreducible characters of with codegree $d\in {\rm Cod}(G)$. We show that there exist two finite $p$-groups with distinct orders that have the same group pseudo-algebra, providing an answer to Question 3.2 in \cite{Moreto2023}. In addition, we also discuss under what hypothesis two $p$-groups with the same group pseudo-algebra will be isomorphic.
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A note on the codegree of finite groups

2024-02-19
Mark L. Lewis , Quanfu Yan
Let $\chi$ be an irreducible character of a group $G,$ and $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi)$ be the sum of the codegrees of the irreducible characters of $G.$ Write ${\rm fcod} … Let $\chi$ be an irreducible character of a group $G,$ and $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi)$ be the sum of the codegrees of the irreducible characters of $G.$ Write ${\rm fcod} (G)=\frac{S_c(G)}{|G|}.$ We aim to explore the structure of finite groups in terms of ${\rm fcod} (G).$ On the other hand, we determine the lower bound of $S_c(G)$ for nonsolvable groups and prove that if $G$ is nonsolvable, then $S_c(G)\geq S_c(A_5)=68,$ with equality if and only if $G\cong A_5.$ Additionally, we show that there is a solvable group so that it has the codegree sum as $A_5.$
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On the sum of character codegrees of finite groups

2024-02-19
Mark L. Lewis , Quanfu Yan
Let $\chi$ be an irreducible character of a group $G.$ We denote the sum of the codegrees of the irreducible characters of $G$ by $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi).$ We consider … Let $\chi$ be an irreducible character of a group $G.$ We denote the sum of the codegrees of the irreducible characters of $G$ by $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi).$ We consider the question if $S_c(G)\leq S_c(C_n)$ is true for any finite group $G,$ where $n=|G|$ and $C_n$ is a cyclic group of order $n.$ We show this inequality holds for many classes of groups. In particular, we provide an affirmative answer for any finite group whose order is divisible by up to 99 primes. However, we show that the question does not hold true in all cases, by evidence of a counterexample.
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Ducci on $\mathbb{Z}_m^3$ and the Max Period

2024-02-12
Mark L. Lewis , Shannon M. Tefft
Let $D(x_1, x_2, ..., x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$ where $D \in End(\mathbb{Z}_m^n)$ be the Ducci function. The sequence $\{D^k(\mathbf{u})\}_{k=0}^{\infty}$ … Let $D(x_1, x_2, ..., x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$ where $D \in End(\mathbb{Z}_m^n)$ be the Ducci function. The sequence $\{D^k(\mathbf{u})\}_{k=0}^{\infty}$ will eventually enter a cycle. If $n=3$, we aim to establish the longest a cycle can be for a given $m$.
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Characterizing finite groups whose enhanced power graphs have universal vertices

2024-02-08
David G. Costanzo , Mark L. Lewis , Stefano Schmidt +2 more
Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and … Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set consisting of the universal vertices of $\Delta(G)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient conditon for $K(G)$ to be nontrivial. We also develop a connection between $\Delta(G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes and the diameter of $\Delta(G)$ is $2$.
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The commuting graph of a solvable 𝐴-group

2024-02-02
Rachel Carleton , Mark L. Lewis
Abstract Let 𝐺 be a finite group. Recall that an 𝐴-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the … Abstract Let 𝐺 be a finite group. Recall that an 𝐴-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable 𝐴-group. Assuming that the commuting graph is connected, we show when the derived length of 𝐺 is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.

The Period of Ducci Cycles on $\mathbb{Z}_{2^l}$ for Tuples of Length $2^k$

2024-01-30
Mark L. Lewis , Shannon M. Tefft
Let the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined as $D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$. … Let the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined as $D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$. In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly applying $D$ will eventually result in $(0,0,...,0)$, as well as establish an upper bound for how many iterations it will take for this to happen.
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The Commuting Graph of a Solvable A-Group

2024-01-30
Rachel Carleton , Mark L. Lewis
Let $G$ be a finite group. Recall that an $A$-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter … Let $G$ be a finite group. Recall that an $A$-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable $A$-group. Assuming that the commuting graph is connected, we show when the derived length of $G$ is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved.
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Finite solvable tidy Groups whose orders are divisible by two primes

2024-01-01
Nicolas F. Beike , Rachel Carleton , David G. Costanzo +4 more
In this paper, we investigate finite solvable tidy groups. We classify the tidy $\{ p, q \}$-groups. Combining this with a previous result, we are able to characterize the finite … In this paper, we investigate finite solvable tidy groups. We classify the tidy $\{ p, q \}$-groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.
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THE AVERAGE OF SOME IRREDUCIBLE CHARACTER DEGREES

2024-01-01
RAMADAN EL SHARIF , Mark L. Lewis

Solvable groups whose monomial, monolithic characters have prime power codegrees

2023-12-01
Xiaoyou Chen , Mark L. Lewis
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FINITE SOLVABLE TIDY GROUPS ARE DETERMINED BY HALL SUBGROUPS WITH TWO PRIMES

2023-07-27
Nicolas F. Beike , Rachel Carleton , David G. Costanzo +4 more
Abstract In this paper, we investigate finite solvable tidy groups. We prove that a solvable group with order divisible by at least two primes is tidy if all of its … Abstract In this paper, we investigate finite solvable tidy groups. We prove that a solvable group with order divisible by at least two primes is tidy if all of its Hall subgroups that are divisible by only two primes are tidy.
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<i>p</i>-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY <i>p</i>-GROUPS

2023-04-20
Nicolas F. Beike , Rachel Carleton , David G. Costanzo +4 more
Abstract Let G be a p -group for some prime p . Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G … Abstract Let G be a p -group for some prime p . Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p . In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p -groups.
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Group Elements Whose Character Values are Roots of Unity

2023-03-08
Mark L. Lewis , Lucia Morotti , Hung P. Tong‐Viet
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Conjugacy classes of maximal cyclic subgroups

2023-02-01
Mariagrazia Bianchi , Rachel D. Camina , Mark L. Lewis +1 more
Abstract In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> … Abstract In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺, denoted <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \eta(G) . First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups 𝑁 with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>N</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \eta(G/N)=\eta(G) . In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:msup> <m:mi>G</m:mi> <m:mo>−</m:mo> </m:msup> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:mrow> </m:math> G/\langle G^{-}\rangle , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>G</m:mi> <m:mo>−</m:mo> </m:msup> </m:math> G^{-} is the set of elements of 𝐺 generating non-maximal cyclic subgroups of 𝐺. More precisely, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">⟨</m:mo> <m:msup> <m:mi>G</m:mi> <m:mo>−</m:mo> </m:msup> <m:mo stretchy="false">⟩</m:mo> </m:mrow> </m:mrow> </m:math> G/\langle G^{-}\rangle is either trivial, elementary abelian, a Frobenius group or isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mn>5</m:mn> </m:msub> </m:math> A_{5} .
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Conjugacy classes of maximal cyclic subgroups of metacyclic 𝑝-groups

2022-10-14
Mariagrazia Bianchi , Rachel D. Camina , Mark L. Lewis
Abstract In this paper, we set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \eta(G) to be the number of conjugacy classes of maximal … Abstract In this paper, we set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \eta(G) to be the number of conjugacy classes of maximal cyclic subgroups of a finite group 𝐺. We compute <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \eta(G) for all metacyclic 𝑝-groups. We show that if 𝐺 is a metacyclic 𝑝-group of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> p^{n} that is not dihedral, generalized quaternion, or semi-dihedral, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>η</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> \eta(G)\geq n-2 , and we determine when equality holds.
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Groups Having all Elements off a Normal Subgroup with Prime Power Order

2022-10-12
Mark L. Lewis
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Toward a classification of the supercharacter theories of <i>C</i><sub><i>p</i></sub> × <i>C</i><sub><i>p</i></sub>

2022-10-12
Shawn T. Burkett , Mark L. Lewis
Abstract In this paper, we study the supercharacter theories of elementary abelian $p$ -groups of order $p^{2}$ . We show that the supercharacter theories that arise from the direct product … Abstract In this paper, we study the supercharacter theories of elementary abelian $p$ -groups of order $p^{2}$ . We show that the supercharacter theories that arise from the direct product construction and the $\ast$ -product construction can be obtained from automorphisms. We also prove that any supercharacter theory of an elementary abelian $p$ -group of order $p^{2}$ that has a non-identity superclass of size $1$ or a non-principal linear supercharacter must come from either a $\ast$ -product or a direct product. Although we are unable to prove results for general primes, we do compute all of the supercharacter theories when $p = 2,\, 3,\, 5$ , and based on these computations along with particular computations for larger primes, we make several conjectures for a general prime $p$ .
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A characterization of GVZ groups in terms of fully ramified characters

2022-06-01
Shawn T. Burkett , Mark L. Lewis
In this paper‎, ‎we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients‎. ‎This characterization is based on counting formulas due to Gallagher‎. In this paper‎, ‎we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients‎. ‎This characterization is based on counting formulas due to Gallagher‎.

CONJUGACY CLASSES OF MAXIMAL CYCLIC SUBGROUPS AND NILPOTENCE CLASS OF -GROUPS

2022-03-21
Mariagrazia Bianchi , Rachel D. Camina , Mark L. Lewis
Abstract Let $\eta (G)$ be the number of conjugacy classes of maximal cyclic subgroups of G . We prove that if G is a p -group of order $p^n$ and … Abstract Let $\eta (G)$ be the number of conjugacy classes of maximal cyclic subgroups of G . We prove that if G is a p -group of order $p^n$ and nilpotence class l , then $\eta (G)$ is bounded below by a linear function in $n/l$ .
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The Rationality of Sylow 2-Subgroups of Solvable $$\mathbb {Q}_{1}$$-Groups

2022-02-22
Mark L. Lewis , M. Norooz-Abadian , Hesam Sharifi

Corrigendum to “Brauer characters of 𝑞′- degree”

2022-02-09
Mark L. Lewis , Hung P. Tong‐Viet
In this corrigendum, we correct some errors in our proofs of part (ii) of Lemma 2.6 and Claim (5) of Theorem 2.7 in [Proc. Amer. Math. Soc. 145 (2017), pp. … In this corrigendum, we correct some errors in our proofs of part (ii) of Lemma 2.6 and Claim (5) of Theorem 2.7 in [Proc. Amer. Math. Soc. 145 (2017), pp. 1891–1898].

Conjugacy classes of maximal cyclic subgroups and nilpotence class of $p$-groups

2022-01-01
M. Bianchi , R. D. Camina , Mark L. Lewis
In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order … In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $\eta (G)$ is bounded below by a linear function in $n/l$.
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Groups having all elements off a normal subgroup with prime power order

2022-01-01
Mark L. Lewis
We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a … We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G \setminus N$ have prime power orders and the orders are divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$ is either a Frobenius group or a $2$-Frobenius group. If all the elements of $G \setminus N$ have prime power orders and the orders are divisible by at least three primes, then all elements of $G$ have prime power order and $G/N$ is nonsolvable.

Conjugacy classes of maximal cyclic subgroups

2022-01-01
M. Bianchi , R. D. Camina , Mark L. Lewis +1 more
In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $\eta$ and direct and semi-direct products. We characterize … In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $\eta$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $\eta (G/N) = \eta (G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomorphic to $A_5$.
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Coauthor Papers Together
Shawn T. Burkett 28
Hung P. Tong‐Viet 16
David G. Costanzo 15
Xiaoyou Chen 14
James P. Cossey 13
Rachel D. Camina 10
Mariagrazia Bianchi 9
Emanuele Pacifici 9
Quanfu Yan 9
D. L. White 8
Rachel Carleton 8
I. M. Isaacs 7
Jamie D. Pearce 7
Nicolas F. Beike 7
A. R. Moghaddamfar 7
Sarah Croome 6
Gabriel Navarro 6
Hung Ngoc Nguyen 6
Kaiwen Lu 6
Colin Heath 6
Shannon M. Tefft 6
Gabe Udell 5
Mark W. Bissler 5
Eyob Tsegaye 5
John K. McVey 5
Stefano Schmidt 5
Lucia Morotti 4
Ni Du 4
Alexander Moretó 4
Lucía Sanus 4
Casey Wynn 4
J. Mirzajani 4
Yanjun Liu 4
M. Bianchi 3
Jonathan Lamar 3
Thomas R. Wolf 3
Qingyun Meng 3
R. D. Camina 3
Thomas Michael Keller 3
A. A. Schaeffer Fry 3
Jacob Laubacher 3
Kamal Aziziheris 3
James Hamblin 3
Carmine Monetta 3
Mehdi Ghaffarzadeh 2
Д. В. Лыткина 2
V. D. Mazurov 2
Josh Hiller 2
Murray Schacher 2
Yiftach Barnea 2

Character theory of finite groups

1999-07-01
I. M. Isaacs
Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur … Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.
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Character Theory of Finite Groups

1976-01-01
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The Magma Algebra System I: The User Language

1997-09-01
Wieb Bosma , John Cannon , Catherine Playoust
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Character Theory of Finite Groups

1998-12-31
Bertram Huppert
Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small … Notations and results from group theory representations and representation-modules simple and semisimple modules orthogonality relations the group algebra characters of abelian groups degrees of irreducible representations characters of some small groups products of representation and characters on the number of solutions gm =1 in a group a theorem of A. Hurwitz on multiplicative sums of squares permutation representations and characters the class number real characters and real representations Coprime action groups pa qb Fronebius groups induced characters Brauer's permutation lemma and Glauberman's character correspondence Clifford theory 1 projective representations Clifford theory 2 extension of characters Degree pattern and group structure monomial groups representation of wreath products characters of p-groups groups with a small number of character degrees linear groups the degree graph groups all of whose character degrees are primes two special degree problems lengths of conjugacy classes R. Brauer's theorem on the character ring applications of Brauer's theorems Artin's induction theorem splitting fields the Schur index integral representations three arithmetical applications small kernels and faithful irreducible characters TI-sets involutions groups whose Sylow-2-subgroups are generalized quaternion groups perfect Fronebius complements. (Part contents).

Characters of Π-separable groups

1984-01-01
I. M. Isaacs

Coprime Group Actions Fixing All Nonlinear Irreducible Characters

1989-02-01
I. M. Isaacs
The main result of this paper is the following: Theorem A. Let H and N be finite groups with coprime orders and suppose that H acts nontrivially on N via … The main result of this paper is the following: Theorem A. Let H and N be finite groups with coprime orders and suppose that H acts nontrivially on N via automorphisms. Assume that H fixes every nonlinear irreducible character of N. Then the derived subgroup of N is nilpotent and so N is solvable of nilpotent length ≦ 2. Why might one be interested in a situation like this? There has been considerable interest in the question of what one can deduce about a group G from a knowledge of the set cd(G) = ﹛x(l)lx ∈ Irr(G) ﹜ of irreducible character degrees of G. Recently, attention has been focused on the prime divisors of the elements of cd(G). For instance, in [9], O. Manz and R. Staszewski consider π-separable groups (for some set π of primes) with the property that every element of cd(G) is either a 77-number or a π'-number.
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Characters of Solvable and Symplectic Groups

1973-01-01
I. M. Isaacs

Solvable groups whose degree graphs have two connected components

2001-01-05
Mark L. Lewis

Generalized Frobenius groups

1984-12-01
David Chillag , I. D. Macdonald

Groups Having Three Complex Irreducible Character Degrees

1995-08-01
Thomas Noritzsch

On the number of components of a graph related to character degrees

1988-01-01
Olaf Manz , Reiner Staszewski , Wolfgang M. Willems
We connect two nonlinear irreducible character of a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if their degrees have a common prime … We connect two nonlinear irreducible character of a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if their degrees have a common prime divisor. In this paper we show that the corresponding graph has at most three connected components.
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Some conditions which almost characterize Frobenius groups

1978-06-01
A. R. Camina

Induction and Restriction of <i>π</i>-Special Characters

1986-06-01
I. M. Isaacs
1. Introduction. The character theory of solvable groups has undergone significant development during the last decade or so and it can now be seen to have quite a rich structure. … 1. Introduction. The character theory of solvable groups has undergone significant development during the last decade or so and it can now be seen to have quite a rich structure. In particular, there is an interesting interaction between characters and sets of prime numbers. Let G be solvable and let π be a set of primes. The “π-special” characters of G are certain irreducible complex characters (defined by D. Gajendragadkar [ 1 ]) which enjoy some remarkable properties, many of which were proved in [ 1 ]. (We shall review the definition and relevant facts in Section 3 of this paper.) Actually, we need not assume solvability: that G is π -separable is sufficient, if we are willing to use the Feit-Thompson “odd order” theorem occasionally. We shall state and prove our results under this weaker hypothesis, but we stress that anything of interest in them is already interesting in the solvable case where, of course, the “odd order” theorem is irrelevant.
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Character degree graphs that are complete graphs

2006-08-31
Mariagrazia Bianchi , David Chillag , Mark L. Lewis +1 more
Let $G$ be a finite group, and write $\operatorname {cd}(G)$ for the set of degrees of irreducible characters of $G$. We define $\Gamma (G)$ to be the graph whose vertex … Let $G$ be a finite group, and write $\operatorname {cd}(G)$ for the set of degrees of irreducible characters of $G$. We define $\Gamma (G)$ to be the graph whose vertex set is $\operatorname {cd}(G)-\{1\}$, and there is an edge between $a$ and $b$ if $(a,b)>1$. We prove that if $\Gamma (G)$ is a complete graph, then $G$ is a solvable group.
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Research in Representation Theory at Mainz (1984–1990)

1991-01-01
Bertram Huppert
This is a report on research conducted at Mainz from 1984 to 1990, made possible by the DFG-project. This is a report on research conducted at Mainz from 1984 to 1990, made possible by the DFG-project.

Somep-groups of Frobenius and extra-special type

1981-09-01
I. D. Macdonald

Non-divisibility among character degrees

2005-01-19
Mark L. Lewis , Alexander Moretó , Thomas R. Wolf
In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the … In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the condition no divisibility among degrees (NDAD). We conjecture that the number of character degrees of a group that satisfies NDAD is bounded and we prove this for solvable groups. More precisely, we prove that solvable groups with NDAD have at most four character degrees and have derived length at most 3. We give a group-theoretic characterization of the solvable groups satisfying NDAD with four character degrees. Since the structure of groups with at most three character degrees is known, these results describe the structure of solvable groups with NDAD.

Characters vanishing on all but two conjugacy classes

1983-12-01
Stephen M. Gagola
If the character table of a group G has a row (corresponding to an irreducible character) with precisely two nonzero entries, then G has a unique minimal normal subgroup N … If the character table of a group G has a row (corresponding to an irreducible character) with precisely two nonzero entries, then G has a unique minimal normal subgroup N which is necessarily an elementary abelian p-grovφ for some prime p.The group G/O p (G) is completely determined here.In general, there is no bound on the derived length or nilpotence class of O p (G). 1.Introduction.An old theorem of Burnside asserts that, for any group G, any irreducible character of degree greater than 1 vanishes at some element of G (for a proof of this fact, see p. 40 of [7]).The extreme case will be considered here, namely, groups G for which a character exists which vanishes on all but two conjugacy classes.Clearly no irreducible character can vanish on all but one conjugacy class (unless \G\ = 1).The remaining sections of this paper are devoted to determining the structure of such groups G. Specifically, §2 is devoted to some preliminary lemmas about the action of G on its unique minimal normal subgroup N. The kernel of G on N is C G (N) = O p (G) for some prime/?and G/O p (G) is determined by Theorems 4.2 and 5.6.The subgroup O p (G) can be quite complicated and this, together with some examples, are discussed in §6. Some preliminary results.As already mentioned in the previous section, if a group Gΐias an irreducible character which does not vanish on only two conjugacy classes, then G has a unique minimal normal subgroup N. The first lemma of this section establishes this, in addition to some properties of the action of G on N.LEMMA 2.1.Let G be a group which has an irreducible character χ such that x does not vanish on exactly two conjugacy classes ofG.If\G\>2 then X is unique and is, moreover, the unique faithful irreducible character of G.In all cases, G contains a unique minimal normal subgroup N which is necessarily an elementary abelian p-group for some prime p.The character χ vanishes on G -N and is nonzero on N. Finally, the action of G by conjugation on N is transitive on N$.Proof.The conclusion of the theorem is trivial if |G| = 2, so assume \G\ > 2. Clearly χ does not vanish at 1 G G.
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Determining Group Structure from Sets of Irreducible Character Degrees

1998-08-01
Mark L. Lewis

Groups with two extreme character degrees and their normal subgroups

2001-02-07
Gustavo A. Fernández‐Alcober , Alexander Moretó
We study the finite groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c d left-parenthesis upper … We study the finite groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c d left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>cd</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {cd}(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of irreducible complex character degrees consists of the two most extreme possible values, that is, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper G colon upper Z left-parenthesis upper G right-parenthesis EndAbsoluteValue Superscript 1 slash 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|G:Z(G)|^{1/2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We are easily reduced to finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups, for which we derive the following group theoretical characterization: they are the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper G colon upper Z left-parenthesis upper G right-parenthesis EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|G:Z(G)|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a square and whose only normal subgroups are those containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G prime"> <mml:semantics> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">G’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or contained in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Z(G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By analogy, we also deal with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper G colon upper Z left-parenthesis upper G right-parenthesis EndAbsoluteValue equals p Superscript 2 n plus 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">|G:Z(G)|=p^{2n+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not a square, and we prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c d left-parenthesis upper G right-parenthesis equals StartSet 1 comma p Superscript n Baseline EndSet"> <mml:semantics> <mml:mrow> <mml:mi>cd</mml:mi> <mml:mo>⁡</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {cd}(G) =\{1,p^n\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if a similar property holds: for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N normal-subgroup-of-or-equal-to upper G"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>⊴</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\trianglelefteq G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G prime less-than-or-equal-to upper N"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>≤</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G’\le N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper N upper Z left-parenthesis upper G right-parenthesis colon upper Z left-parenthesis upper G right-parenthesis EndAbsoluteValue less-than-or-equal-to p"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">|NZ(G):Z(G)|\le p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof of these results requires a detailed analysis of the structure of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the index of the centre is small, and in some cases we may even bound the order of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

The vanishing-off subgroup

2008-12-19
Mark L. Lewis
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On Camina Groups of Prime Power Order

1996-05-01
Rex Dark , Carlo Maria Scoppola

An Overview of Graphs Associated with Character Degrees And Conjugacy Class Sizes in Finite Groups

2008-02-01
Mark L. Lewis
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Connectedness of degree graphs of nonsolvable groups

2003-06-30
Mark L. Lewis , D. L. White

Bounding Fitting Heights of Character Degree Graphs

2001-08-01
Mark L. Lewis

Irreducible Character Degrees and Normal Subgroups

1998-01-01
I. M. Isaacs , Greg Knutson
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ISOMORPHIC CHARACTER TABLES OF NESTED <i>GVZ</i>-GROUPS

2011-05-31
Adriana Nenciu
In this paper we introduce the notion of nested GVZ-groups. We give necessary and sufficient conditions for two non-isomorphic nested GVZ-groups to have isomorphic character tables. In this paper we introduce the notion of nested GVZ-groups. We give necessary and sufficient conditions for two non-isomorphic nested GVZ-groups to have isomorphic character tables.

Solvable Groups Having Almost Relatively Prime Distinct Irreducible Character Degrees

1995-05-01
Mark L. Lewis

On a Class of Doubly Transitive Groups

1962-01-01
Michio Suzuki

More onp-groups of Frobenius type

1986-10-01
I. D. Macdonald

Partial characters of π-separable groups

1991-01-01
I. M. Isaacs
Our concern in this expository paper is the character theory of a finite group G as seen from the perspective of a set π of prime numbers. There are no … Our concern in this expository paper is the character theory of a finite group G as seen from the perspective of a set π of prime numbers. There are no new results here and few new ideas. Our purpose is to present in as accessible a manner as possible, the proofs of some theorems in the character theory of π-separable groups, and to explain the significance of these results.

A characteristic class of characters of finite π-separable groups

1979-08-01
Dilip Gajendragadkar

Nested GVZ-groups

2016-01-08
Adriana Nenciu
Abstract We investigate the finite groups G for which <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>=</m:mo> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>Z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> … Abstract We investigate the finite groups G for which <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>=</m:mo> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>Z</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:math> ${\chi(1)^{2}=|G:Z(\chi)|}$ for all characters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>χ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>Irr</m:mo> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> ${\chi\in\operatorname{Irr}(G)}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>Z</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mi>χ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>Irr</m:mo> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> ${\{Z(\chi):\chi\in\operatorname{Irr}(G)\}}$ is a chain with respect to inclusion. We call these groups nested GVZ-groups. We will show that for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> ${n\geq 2}$ there is a nested GVZ-group with nilpotency class n .

On a Class of Doubly Transitive Groups: II

1964-05-01
Michio Suzuki
Abstract : This study considers a class of doubly transitive groups satisfying the condition that the identity is the only element leaving three distinct letters fixed. The main object of … Abstract : This study considers a class of doubly transitive groups satisfying the condition that the identity is the only element leaving three distinct letters fixed. The main object of the investigation is to classify the groups which do not contain a regular normal subgroup of order 1 + N in case N is even. (Author)

Bounds on the number of lifts of a Brauer character in a p-solvable group

2007-03-27
James P. Cossey
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Fong characters in π-separable groups

1986-03-01
I. M. Isaacs

Semi-extraspeziellep-Gruppen

1977-10-01
Bert Beisiegel

Groups of Prime Power Order 4

2015-12-11
Yakov Berkovich , Zvonimir Janko
This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p- groups play an important role. Topics covered … This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p- groups play an important role. Topics covered in this volume include: subgroup structure of metacyclic p- groups Ishikawa's theorem on p- groups with two sizes of conjugate classes p- central p- groups theorem of Kegel on nilpotence of H p -groups partitions of p- groups characterizations of Dedekindian groups norm of p- groups p- groups with 2-uniserial subgroups of small order The book also contains hundreds of original exercises and solutions and a comprehensive list of more than 500 open problems. This work is suitable for researchers and graduate students with a modest background in algebra.

Gruppi semiextraspeciali di esponentep

1987-12-01
Libero Verardi
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Supercharacters and superclasses for algebra groups

2007-11-20
Persi Diaconis , I. M. Isaacs
We study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups. These groups generalize the unimodular upper triangular groups over a finite field, and … We study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups. These groups generalize the unimodular upper triangular groups over a finite field, and the supercharacter theory we develop extends results of Carlos André and Ning Yan that were originally proved in the upper triangular case. This theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table. We discuss connections with the Kirillov orbit method and with Gelfand pairs, and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class, respectively. We also show that products of supercharacters are positive integer combinations of supercharacters.

Supercharacter Theory Constructions Corresponding to Schur Ring Products

2012-10-10
Anders O. F. Hendrickson
Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs … Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs , I. M. ( 2008 ). Supercharacters and superclasses for algebra groups . Trans. Amer. Math. Soc. 360 : 2359 – 2392 .[Crossref], [Web of Science ®] , [Google Scholar]]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12 Leung , K. H. , Man , S. H. ( 1996 ). On Schur rings over cyclic groups, II . J. Algebra 183 : 273 – 285 .[Crossref], [Web of Science ®] , [Google Scholar]], Hirasaka and Muzychuk [10 Hirasaka , M. , Muzychuk , M. ( 2001 ). An elementary abelian group of rank 4 is a CI-group . J. Combin. Theory Ser. A 94 : 339 – 362 .[Crossref], [Web of Science ®] , [Google Scholar]], and Tamaschke [20 Tamaschke , O. ( 1970 ). On Schur-rings which define a proper character theory on finite groups . Math. Z. 117 : 340 – 360 .[Crossref], [Web of Science ®] , [Google Scholar]].

Codegrees and nilpotence class of <i>p</i>-groups

2015-12-17
Ni Du , Mark L. Lewis
Abstract If χ is an irreducible character of a finite group G , then the codegree of χ is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> … Abstract If χ is an irreducible character of a finite group G , then the codegree of χ is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>|</m:mo> <m:mi>G</m:mi> <m:mo>:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>χ</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>|</m:mo> <m:mo>/</m:mo> <m:mi>χ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> ${|G:{\rm ker}(\chi)|/\chi(1)}$ . We show that if G is a p -group, then the nilpotence class of G is bounded in terms of the largest codegree for an irreducible character of G .

Groups where the centers of the irreducible characters form a chain

2020-01-01
Mark L. Lewis
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Extending the Steinberg representation

1992-08-01
Peter Schmid

Characters and Hall Subgroups of Groups of Odd Order

1993-06-01
I. M. Isaacs

Coprimeness among irreducible character degrees of finite solvable groups

1997-01-01
Diane Benjamin
Given a finite solvable group $G$, we say that $G$ has property $P_{k}$ if every set of $k$ distinct irreducible character degrees of $G$ is (setwise) relatively prime. Let $k(G)$ … Given a finite solvable group $G$, we say that $G$ has property $P_{k}$ if every set of $k$ distinct irreducible character degrees of $G$ is (setwise) relatively prime. Let $k(G)$ be the smallest positive integer such that $G$ satisfies property $P_{k}$. We derive a bound, which is quadratic in $k(G)$, for the total number of irreducible character degrees of $G$. Three exceptional cases occur; examples are constructed which verify the sharpness of the bound in each of these special cases.

DERIVED LENGTHS OF SOLVABLE GROUPS SATISFYING THE ONE-PRIME HYPOTHESIS. III

2001-01-01
Mark L. Lewis
A finite group G is said to satisfy the one-prime hypothesis if the greatest common divisor of any two distinct irreducible character degrees is either 1 or a prime number. … A finite group G is said to satisfy the one-prime hypothesis if the greatest common divisor of any two distinct irreducible character degrees is either 1 or a prime number. The principal result of this paper is that if G is solvable and satisfies the one-prime hypothesis, and if G has a nonabelian nilpotent homomorphic image, then the derived length of G is at most 4.

Algebra, a graduate course

1995-01-01
I. M. Isaacs
This book, based on a first-year graduate course the author taught at the University of Wisconsin, contains more than enough material for a two-semester graduate-level abstract algebra course, including groups, … This book, based on a first-year graduate course the author taught at the University of Wisconsin, contains more than enough material for a two-semester graduate-level abstract algebra course, including groups, rings and modules, fields and Galois theory, an introduction to algebraic number theory, and the rudiments of algebraic geometry. In addition, there are some more specialized topics not usually covered in such a course. These include transfer and character theory of finite groups, modules over artinian rings, modules over Dedekind domains, and transcendental field extensions. This book could be used for self study as well as for a course text, and so full details of almost all proofs are included, with nothing being relegated to the chapter-end problems. There are, however, hundreds of problems, many being far from trivial. The book attempts to capture some of the informality of the classroom, as well as the excitement the author felt when taking the corresponding course as a student.

A characterization of groups in terms of the degrees of their characters

1965-09-01
I. M. Isaacs , D. S. Passman
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SOLVABLE GROUPS WITH CHARACTER DEGREE GRAPHS HAVING 5 VERTICES AND DIAMETER 3

2002-12-11
Mark L. Lewis
ABSTRACT Let G be a finite group and cd(G) the character degrees of G. The degree graph Δ(G) of G is the graph whose vertices are the primes dividing degrees … ABSTRACT Let G be a finite group and cd(G) the character degrees of G. The degree graph Δ(G) of G is the graph whose vertices are the primes dividing degrees in cd(G), and there is an edge between p and q if pq divides some degree in cd(G). In this paper, we show that if Δ(G) has 5 vertices, then the diameter of Δ(G) is at most 2. This shows that the example in[9] of a solvable group G where Δ(G) has diameter 3 has the fewest number of vertices possible.