Pálfy proved that given a solvable group G and a set of prime divisors of character degrees of G of cardinality at least 3, there exist two different primes such …
Pálfy proved that given a solvable group G and a set of prime divisors of character degrees of G of cardinality at least 3, there exist two different primes such that pq divides some character degree. The solvability hypothesis cannot be removed from Pálfy's theorem, but we show that the same conclusion holds for arbitrary finite groups if .
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>.
We prove that if the average of the degrees of the irreducible characters of a finite group G is less than 16 5 , then G is solvable. This solves …
We prove that if the average of the degrees of the irreducible characters of a finite group G is less than 16 5 , then G is solvable. This solves a conjecture of I. M. Isaacs, M. Loukaki and the first author. We discuss related questions.
Journal Article Extending Brauer's Height Zero Conjecture to Blocks with Nonabelian Defect Groups Get access Charles W. Eaton, Charles W. Eaton 1School of Mathematics, Alan Turing Building, University of Manchester, …
Journal Article Extending Brauer's Height Zero Conjecture to Blocks with Nonabelian Defect Groups Get access Charles W. Eaton, Charles W. Eaton 1School of Mathematics, Alan Turing Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK Correspondence to be sent to: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Alexander Moretó Alexander Moretó 2Departamento de Algebra, Facultad de Matematicas, Universidad de Valencia, 46100 Burjassot (Valencia), Spain Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2014, Issue 20, 2014, Pages 5581–5601, https://doi.org/10.1093/imrn/rnt131 Published: 03 July 2013 Article history Received: 07 March 2013 Revision received: 29 May 2013 Accepted: 04 June 2013 Published: 03 July 2013
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups that can easily be verified using the character table.
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups that can easily be verified using the character table.
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.
Let $G$ be a finite group that acts on a nonzero finite dimensional vector space $V$ over an arbitrary field. Assume that $V$ is completely reducible as a $G$-module, and …
Let $G$ be a finite group that acts on a nonzero finite dimensional vector space $V$ over an arbitrary field. Assume that $V$ is completely reducible as a $G$-module, and that $G$ fixes no nonzero vector of $V$. We show that some element $g\in G$ has a small fixed-point space in $V$. Specifically, we prove that we can choose $g$ so that $\dim \mathbf {C}_V(g)\le (1/p)\dim V$, where $p$ is the smallest prime divisor of $|G|$.
The Ito-Michler theorem asserts that if no irreducible character of a finite group $G$ has degree divisible by some given prime $p$, then a Sylow $p$-subgroup of $G$ is both …
The Ito-Michler theorem asserts that if no irreducible character of a finite group $G$ has degree divisible by some given prime $p$, then a Sylow $p$-subgroup of $G$ is both normal and abelian. In this paper we relax the hypothesis, and we assume that there is at exactly one multiple of $p$ that occurs as the degree of an irreducible character of $G$. We show that in this situation, a Sylow $p$-subgroup of $G$ is almost normal in $G$, and it is almost abelian.
Let p be a prime. The goal of this paper is to classify the finite groups with exactly one conjugacy class of size a multiple of p.
Let p be a prime. The goal of this paper is to classify the finite groups with exactly one conjugacy class of size a multiple of p.
The aim of this note is to present some problems and also partial results in some cases, mainly on characters of p-groups. (In the last section we deal with a …
The aim of this note is to present some problems and also partial results in some cases, mainly on characters of p-groups. (In the last section we deal with a problem that consists in obtaining information about characters of a Sylow p-subgroup of an arbitrary group from information about the characters of the whole group.) This survey is far from being exhaustive. The topics included are strongly influenced by the author's interests in the last few years. There seems to be an increasing interest in the character theory of p-groups and we hope that this expository paper will encourage more research in the area. In the sixties I. M. Isaacs and D. S. Passman [17, 18] wrote two important papers that initiated the study of the degrees of the irreducible complex characters of finite groups (henceforth referred to as character degrees). The study of the influence of the set of character degrees on the structure of a group was taken up again in the eighties, in large part due to B. Huppert and his school. In particular, this has led to several papers dealing with the character degrees of important families of p-groups since the nineties (see [6, 8, 12, 28, 30, 32, 33, 34, 35, 36, 37]). Here we are mostly concerned with character degrees, but instead of studying particular families of p-groups, we intend to obtain general structural properties of groups according to their character degrees. Other problems on characters of p-groups appear in [25].
In this paper, it is proved that if B is a Brauer p-block of a p-solvable group, for some odd prime p, then the height of any ordinary character in …
In this paper, it is proved that if B is a Brauer p-block of a p-solvable group, for some odd prime p, then the height of any ordinary character in B is at most 2b, where pb is the largest degree of the irreducible characters of the defect group of B. Some other results that relate the heights of characters with properties of the defect group are obtained. 2000 Mathematics Subject Classification 20C15, 20C20.
In this note we prove that for any two integers $r,s>1$ there exist finite $p$-groups $G$ of class $2$ such that $|\operatorname {cd}(G)|=r$ and $|\operatorname {cs}(G)|=s$.
In this note we prove that for any two integers $r,s>1$ there exist finite $p$-groups $G$ of class $2$ such that $|\operatorname {cd}(G)|=r$ and $|\operatorname {cs}(G)|=s$.
We prove that the derived length of a solvable group is bounded in terms of certain invariants associated to the set of character degrees and improve some of the known …
We prove that the derived length of a solvable group is bounded in terms of certain invariants associated to the set of character degrees and improve some of the known bounds. We also bound the derived length of a Sylow $p$-subgroup of a solvable group by the number of different $p$-parts of the character degrees of the whole group.
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>be a finite set of powers of<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>containing 1. It is known that for some choices of<inline-formula …
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>be a finite set of powers of<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>containing 1. It is known that for some choices of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>, if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>is a 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>-group whose set of character degrees is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>, then the nilpotence class of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"><mml:semantics><mml:mi>P</mml:mi><mml:annotation encoding="application/x-tex">P</mml:annotation></mml:semantics></mml:math></inline-formula>is bounded by some integer that depends on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>, while for some other choices of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>is class bounding if and only if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p not-an-element-of script upper S"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>∉</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">p\notin \mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>. In this article we provide a new approach to this problem. Our main result shows the relevance of certain<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>-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>such that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p not-an-element-of script upper S"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>∉</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">S</mml:mi></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">p\notin \mathcal {S}</mml:annotation></mml:semantics></mml:math></inline-formula>.
We say that a finite group G is an NDAD-group (no divisibility among degrees) if for any 1 < a < b in the set of degrees of the complex …
We say that a finite group G is an NDAD-group (no divisibility among degrees) if for any 1 < a < b in the set of degrees of the complex irreducible characters of G, a does not divide b. In this article, we determine the nonsolvable NDAD-groups. Together with the work of Lewis, Moretó and Wolf (J. Group Theory 8 (2005)), this settles a problem raised by Berkovich and Zhmud’, which asks for a classification of the NDAD-groups.
Let G be a finite group, let p be a prime and let Pr p (G) be the probability that two random p-elements of G commute.In this paper we prove …
Let G be a finite group, let p be a prime and let Pr p (G) be the probability that two random p-elements of G commute.In this paper we prove that Pr p (G) > ( p 2 + p -1)/ p 3 if and only if G has a normal and abelian Sylow p-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group.This bound is best possible in the sense that for each prime p there are groups with Pr p (G) = ( p 2 + p -1)/ p 3 and we classify all such groups.Our proof is based on bounding the proportion of p-elements in G that commute with a fixed p-element in G \ O p (G), which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.
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.
Abstract Baumslag and Wiegold have recently proven that a finite group G is nilpotent if and only if o ( xy ) = o ( x ) o ( y …
Abstract Baumslag and Wiegold have recently proven that a finite group G is nilpotent if and only if o ( xy ) = o ( x ) o ( y ) for every x , y ∈ G with ( o ( x ), o ( y )) = 1. Motivated by this surprisingly new result, we have obtained related results that just consider sets of prime divisors of element orders. For instance, the first of our main results asserts that G is nilpotent if and only if π( o ( xy )) = π( o ( x ) o ( y )) for every x , y ∈ G of prime power order with ( o ( x ), o ( y )) = 1. As an immediate consequence, we recover the Baumslag–Wiegold Theorem. While this result is still elementary, we also obtain local versions that, for instance, characterize the existence of a normal Sylow p -subgroup in terms of sets of prime divisors of element orders. These results are deeper and our proofs rely on results that depend on the classification of finite simple groups.
We prove that the number of different prime divisors of the order of a finite group is bounded by a polynomial function of the maximum of the number of different …
We prove that the number of different prime divisors of the order of a finite group is bounded by a polynomial function of the maximum of the number of different prime divisors of the element orders. This improves a result of J. Zhang.
Brauer’s Problem 1 asks the following: what are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible …
Brauer’s Problem 1 asks the following: what are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to announce a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of isomorphic summands, then its dimension is bounded in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We prove that this is true for every finite group if it is true for the symmetric groups.
Abstract Recently, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if for every of coprime order. Motivated by this result, we study the groups …
Abstract Recently, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if for every of coprime order. Motivated by this result, we study the groups with the property that and those with the property that for every and every nontrivial of pairwise coprime order. We also consider several ways of weakening the hypothesis on x and y . While the result of Baumslag and Wiegold is completely elementary, some of our arguments here depend on (parts of) the classification of finite simple groups.
We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the largest multiplicity of the fields of values of the irreducible characters of …
We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the largest multiplicity of the fields of values of the irreducible characters 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>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper G 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:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|G|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded from above in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
If G is a finite group, p is a prime, and x ∈ G , it is an interesting problem to place x in a convenient small (normal) subgroup of …
If G is a finite group, p is a prime, and x ∈ G , it is an interesting problem to place x in a convenient small (normal) subgroup of G, assuming some knowledge of the order of the products x y , for certain p-elements y of G.
Abstract We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that …
Abstract We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that for any prime p the number of character codegrees of a finite group G that are divisible by p is at most k , then the number of prime divisors of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> {|G|} is bounded in terms of k . We prove this conjecture for solvable groups.
Huppert’s ρ-σ conjecture asserts that any finite group has some character degree that is divisible by “many” primes. In this note, we consider a dual version of this problem, and …
Huppert’s ρ-σ conjecture asserts that any finite group has some character degree that is divisible by “many” primes. In this note, we consider a dual version of this problem, and we prove that for any finite group there is some prime that divides “many” character degrees.
We classify finite groups with a small average number of zeros in the character table.
We classify finite groups with a small average number of zeros in the character table.
The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group …
The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a $2$-group with elementary abelian center of order $8$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ are fully ramified with respect to $G/Z(G)$. We also obtain bounds for the representation dimension of quotients of $G$ in terms of the representation dimension of $G$, and discuss the relation of this invariant with the essential dimension of $G$.
Let G $G$ be a finite group. We show that the order of a finite group is bounded above in terms of the largest multiplicity of the character codegrees.
Let G $G$ be a finite group. We show that the order of a finite group is bounded above in terms of the largest multiplicity of the character codegrees.
Abstract Fixed‐point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime …
Abstract Fixed‐point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime and a finite group , we use fixed‐point ratios to study the number of Sylow ‐subgroups of and the minimal size of a covering by proper subgroups of the set of ‐elements of .
Abstract Recently, Malle and Navarro provided a Galois-theoretic enhancement of Brauer’s height zero conjecture for principal $p$-blocks, limited to the case $p=2$, utilizing a specific Galois automorphism of order 2. …
Abstract Recently, Malle and Navarro provided a Galois-theoretic enhancement of Brauer’s height zero conjecture for principal $p$-blocks, limited to the case $p=2$, utilizing a specific Galois automorphism of order 2. While they left open the question of whether a similar result could hold for odd primes, in this paper, we significantly advance their work by formulating a broader Galois version of the conjecture for any prime $p$, using an elementary abelian $p$-subgroup of the absolute Galois group. We not only strengthen their result for $p=2$, but also prove the conjecture for arbitrary primes $p$, except when $G$ contains certain small-rank Lie-type groups as composition factors. Moreover, we establish the conjecture for almost simple groups and for $p$-solvable groups.
Problem 21 of Brauer’s list of problems from 1963 asks whether for any positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are finitely …
Problem 21 of Brauer’s list of problems from 1963 asks whether for any positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> there are finitely many isomorphism classes of groups that occur as the defect group of a block with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> irreducible characters is necessarily the cyclic group of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In most cases, we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer’s height zero conjecture.
ABSTRACT Let H be a nonabelian finite simple group. Huppert’s conjecture asserts that if G is a finite group with the same set of complex character degrees as H, then …
ABSTRACT Let H be a nonabelian finite simple group. Huppert’s conjecture asserts that if G is a finite group with the same set of complex character degrees as H, then $G\cong H\times A$ for some abelian group A. Over the past two decades, several specific cases of this conjecture have been addressed. Recently, attention has shifted to the analogous conjecture for character codegrees: if G has the same set of character codegrees as H, then $G\cong H$. Unfortunately, both problems have primarily been examined on a case-by-case basis. In this paper and the companion [15], we present a more unified approach to the codegree conjecture and confirm it for several families of simple groups.
Abstract Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G , …
Abstract Let G be a finite solvable group. We prove that if $\chi\in{{\operatorname{Irr}}}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the nonlinear irreducible characters of G , then $G/\operatorname{Ker}\chi$ is nilpotent-by-abelian.
Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime …
Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.
Abstract We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. …
Abstract We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin–Tits’ simple order theorem.
Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts that if $G$ is a finite group with the same set of complex character degrees as $H$, then $G\cong …
Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts that if $G$ is a finite group with the same set of complex character degrees as $H$, then $G\cong H\times A$ for some abelian group $A$. Over the past two decades, several specific cases of this conjecture have been addressed. Recently, attention has shifted to the analogous conjecture for character codegrees: if $G$ has the same set of character codegrees as $H$, then $G\cong H$. Unfortunately, both problems have primarily been examined on a case-by-case basis. In this paper and the companion [HM22], we present a more unified approach to the codegree conjecture and confirm it for several families of simple groups.
The celebrated It\^o-Michler theorem asserts that a prime $p$ does not divide the degree of any irreducible character of a finite group $G$ if and only if $G$ has a …
The celebrated It\^o-Michler theorem asserts that a prime $p$ does not divide the degree of any irreducible character of a finite group $G$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup. The principal block case of the recently-proven Brauer's height zero conjecture isolates the abelian part in the It\^o-Michler theorem. In this paper, we show that the normal part can also be isolated in a similar way. This is a consequence of work on a strong form of the so-called Brauer's height zero conjecture for two primes of Malle and Navarro. Using our techniques, we also provide an alternate proof of this conjecture.
Conjecture A of [3] predicts the equality between the smallest positive height of the irreducible characters in a p-block of a finite group and the smallest positive height of the …
Conjecture A of [3] predicts the equality between the smallest positive height of the irreducible characters in a p-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any …
Recently, Malle and Navarro obtained a Galois strengthening of Brauer's height zero conjecture for principal $p$-blocks when $p=2$, considering a particular Galois automorphism of order~$2$. In this paper, for any prime $p$ we consider a certain elementary abelian $p$-subgroup of the Galois group and propose a Galois version of Brauer's height zero conjecture for principal $p$-blocks. We prove it when $p=2$ and also for arbitrary $p$ when $G$ does not involve certain groups of Lie type of small rank as composition factors. Furthermore, we prove it for almost simple groups and for $p$-solvable groups.
Abstract We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$ , it is proved that, with one exception, any two irreducible characters …
Abstract We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $\mathsf {S}_n$ , it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group G , with nonlinear irreducible characters of G as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by three. Lastly, the result for $\mathsf {S}_n$ is applied to prove the nonequivalence of the metrics on permutations induced from faithful irreducible characters of the group.
Let Gp be the set of p-elements of a finite group G. Do we need all the Sylow p-subgroups of G to cover Gp? Although this question does not have …
Let Gp be the set of p-elements of a finite group G. Do we need all the Sylow p-subgroups of G to cover Gp? Although this question does not have an affirmative answer in general, our work indicates that the answer is yes more often than one could perhaps expect.
Abstract Fix a prime p and an integer $$n\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Among the non-linear irreducible characters of the p -groups of order …
Abstract Fix a prime p and an integer $$n\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Among the non-linear irreducible characters of the p -groups of order $$p^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , what is the minimum number of elements that take the value 0?
Abstract We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G …
Abstract We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G , then the number of conjugacy classes of G is at least $Dp/\log_2p$ . We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$ .
Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a …
Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.
Let G be a finite group, let p be a prime and let Pr p (G) be the probability that two random p-elements of G commute.In this paper we prove …
Let G be a finite group, let p be a prime and let Pr p (G) be the probability that two random p-elements of G commute.In this paper we prove that Pr p (G) > ( p 2 + p -1)/ p 3 if and only if G has a normal and abelian Sylow p-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group.This bound is best possible in the sense that for each prime p there are groups with Pr p (G) = ( p 2 + p -1)/ p 3 and we classify all such groups.Our proof is based on bounding the proportion of p-elements in G that commute with a fixed p-element in G \ O p (G), which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.
Let K be a field of characteristic p>0. We prove that if all irreducible representations over K of a finite group G have degree at most n, then G has …
Let K be a field of characteristic p>0. We prove that if all irreducible representations over K of a finite group G have degree at most n, then G has a characteristic subgroup N of index bounded above in terms of n such that its derived subgroup N′ is a p-group. Our proof of this result relies on the celebrated Larsen-Pink theorem. We also show that every finite group G has a solvable p-nilpotent subgroup of index bounded above in terms of the largest p′-degree of the complex irreducible characters of G.
If G is a finite group and x∈G, we say that x lies in a small class if |xG| is minimal among the sizes of the noncentral conjugacy classes of …
If G is a finite group and x∈G, we say that x lies in a small class if |xG| is minimal among the sizes of the noncentral conjugacy classes of G. It has been conjectured that if G is a solvable group with trivial center and x belongs to a small class, then x lies in the center of the Fitting subgroup of G. We restrict the structure of a possible counterexample to this conjecture. We discuss the possible existence of a counterexample. As a consequence, we prove the conjecture when the small classes have prime sizes and also when all chief factors of G have rank at most 2. Perhaps surprisingly, the proof in the case when the small classes have prime size and the discussion on the existence of counterexamples use techniques from linear algebra.
Abstract We prove that the order of a finite group 𝐺 with trivial solvable radical is bounded above in terms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>acd</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> …
Abstract We prove that the order of a finite group 𝐺 with trivial solvable radical is bounded above in terms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>acd</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> \operatorname{acd}(G) , the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>acd</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> \operatorname{acd}(G) , but we show that, in certain cases, it is bounded in terms of the degrees of the irreducible characters of 𝐺 that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck’s conjecture.
We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The …
We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin-Tits' simple order theorem.
Fix a prime $p$ and an integer $n\geq 0$. Among the non-linear irreducible characters of the $p$-groups of order $p^n$, what is the minimum number of elements that take the …
Fix a prime $p$ and an integer $n\geq 0$. Among the non-linear irreducible characters of the $p$-groups of order $p^n$, what is the minimum number of elements that take the value 0?
We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least …
We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this phenomenon, we introduce the common-zero graph of a finite group $G$, with non-linear irreducible characters of $G$ as vertices, and edges connecting characters that vanish on some common group element. We show that for solvable and simple groups, the number of connected components of this graph is bounded above by 3. Lastly, the result for $S_n$ is applied to prove the non-equivalence of the metrics on permutations induced from faithful irreducible characters of the group.
Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the …
Conjecture A of \cite{EM14} predicts the equality between the smallest positive height of the irreducible characters in a $p$-block of a finite group and the smallest positive height of the irreducible characters in its defect group. Hence, it can be seen as a generalization of Brauer's famous height zero conjecture. One inequality was shown to be a consequence of Dade's Projective Conjecture. We prove the other, less well understood, inequality for principal blocks when the defect group has two character degrees.
Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group …
Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem for principal blocks. Another long-standing open problem (from 1982) in this area asks whether the defect group of a block with 3 irreducible characters is necessarily the cyclic group of order 3. In most cases we reduce this problem to a question on simple groups that is closely related to the recent solution of Brauer's height zero conjecture.
Let $G$ be a finite solvable group. We prove that if $\chi\in{\rm Irr}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the non-linear irreducible characters of $G$, then …
Let $G$ be a finite solvable group. We prove that if $\chi\in{\rm Irr}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the non-linear irreducible characters of $G$, then $G/{\rm Ker} \chi$ is nilpotent-by-abelian.
Let $G$ be a finite group. We study the generalized character defined by $\Xi(g)=|G|o(g)$, for $g\in G$, which is closely related to a function that has been very studied recently …
Let $G$ be a finite group. We study the generalized character defined by $\Xi(g)=|G|o(g)$, for $g\in G$, which is closely related to a function that has been very studied recently from a group theoretical point of view.
Let G $G$ be a finite group. We show that the order of a finite group is bounded above in terms of the largest multiplicity of the character codegrees.
Let G $G$ be a finite group. We show that the order of a finite group is bounded above in terms of the largest multiplicity of the character codegrees.
Let b(G) be the largest degree of the irreducible characters of a nonabelian finite group G and let m(G) be the smallest degree of the nonlinear irreducible characters of G. …
Let b(G) be the largest degree of the irreducible characters of a nonabelian finite group G and let m(G) be the smallest degree of the nonlinear irreducible characters of G. M. Isaacs defined the character degree ratio of G to be rat(G)=b(G)/m(G). He proved that for nonabelian solvable groups the derived length is bounded by a logarithmic function of the character degree ratio. In a similar spirit, results that restrict the structure of nonsolvable groups in terms of the character degree ratio were later proved by J. P. Cossey, M. Lewis and H. N. Nguyen. In this note we show that, with some rather precisely determined exceptions, G has an abelian subgroup of index bounded by a polynomial function in rat(G). As a consequence we get stronger (in most situations) versions of most of the previously known results on the character degree ratio.
Abstract We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that …
Abstract We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if k is a positive integer such that for any prime p the number of character codegrees of a finite group G that are divisible by p is at most k , then the number of prime divisors of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> {|G|} is bounded in terms of k . We prove this conjecture for solvable groups.
It has been conjectured that if the number of distinct irreducible constituents of the product of two faithful irreducible characters of a finite $p$-group, for $p\geq5$, is bigger than $(p+1)/2$, …
It has been conjectured that if the number of distinct irreducible constituents of the product of two faithful irreducible characters of a finite $p$-group, for $p\geq5$, is bigger than $(p+1)/2$, then it is at least $p$. We give a counterexample to this conjecture.
Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a …
Let G be a finite group with Sylow p-subgroup P. We show that the character table of G determines whether P has maximal nilpotency class and whether P is a minimal non-abelian group. The latter result is obtained from a precise classification of the corresponding groups G in terms of their composition factors. For p-constrained groups G we prove further that the character table determines whether P can be generated by two elements.
We present some variations on some of the main open problems on character degrees. We collect some of the methods that have proven to be very useful to work on …
We present some variations on some of the main open problems on character degrees. We collect some of the methods that have proven to be very useful to work on these problems. These methods are also useful to solve certain problems on zeros of characters, character kernels and fields of values of characters.
We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of ${\rm acd}(G)$, the average degree of the irreducible characters. It …
We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of ${\rm acd}(G)$, the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of ${\rm acd}(G)$, but we show that in certain cases it is bounded in terms of the degrees of the irreducible characters of $G$ that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck's conjecture.
Motivated by a question of A. Bächle, we prove that if the field of values of any irreducible character of a finite group G is imaginary quadratic or rational, then …
Motivated by a question of A. Bächle, we prove that if the field of values of any irreducible character of a finite group G is imaginary quadratic or rational, then the field generated by the character table Q(G)/Q is an extension of degree bounded in terms of the largest alternating group that appears as a composition factor of G. In order to prove this result, we extend a theorem of J. Tent on quadratic rational solvable groups to nonsolvable groups.
We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if $k$ is a positive integer such that for …
We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if $k$ is a positive integer such that for any prime $p$ the number of character codegrees of a finite group $G$ that are divisible by $p$ is at most $k$, then the number of prime divisors of $|G|$ is bounded in terms of $k$. We prove this conjecture for solvable groups.
We classify finite groups with a small average number of zeros in the character table.
We classify finite groups with a small average number of zeros in the character table.
Du and Lewis raised in 2016 the question of whether the nilpotence class of a p-group is bounded in terms of the number of character codegrees. In 2020, Croome and …
Du and Lewis raised in 2016 the question of whether the nilpotence class of a p-group is bounded in terms of the number of character codegrees. In 2020, Croome and Lewis, gave a positive answer to this question for p-groups with four character codegree under some additional hypotheses related, for instance, to the number of character degrees of the group. In this note, we show that in general the nilpotence class of a p-group is bounded in terms of the number of character degrees and the number of character codegrees. In the case of four character codegrees, we extend some of the results of Croome and Lewis.
Abstract We classify finite groups with a small average number of zeros in the character table.
Abstract We classify finite groups with a small average number of zeros in the character table.
We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the largest multiplicity of the fields of values of the irreducible characters of …
We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the largest multiplicity of the fields of values of the irreducible characters 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>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper G 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:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|G|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded from above in terms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group …
The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a $2$-group with elementary abelian center of order $8$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ are fully ramified with respect to $G/Z(G)$. We also obtain bounds for the representation dimension of quotients of $G$ in terms of the representation dimension of $G$, and discuss the relation of this invariant with the essential dimension of $G$.
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.
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).
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a <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>-block with defect 0, completing …
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a <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>-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p minus"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>blocks remained unclassified were the alternating groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript n"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">A_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here we show that these all have a <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>-block with defect 0 for every prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 5"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\geq 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This follows from proving the same result for every symmetric group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which in turn follows as a consequence of the <italic><inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-core partition conjecture</italic>, that every non-negative integer possesses at least one <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-core partition, for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than-or-equal-to 17"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>17</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t\geq 17</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we reduce this problem to Lagrange’s Theorem that every non-negative integer can be written as the sum of four squares. The only case with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t greater-than 17"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>17</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t>17</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that was not covered in previous work, was the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t equals 13"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>13</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">t=13</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This we prove with a very different argument, by interpreting the generating function for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne’s Theorem (née the <italic>Weil Conjectures</italic>). We also consider congruences for the number of <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>-blocks of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, proving a conjecture of Garvan, that establishes certain multiplicative congruences when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="5 less-than-or-equal-to p less-than-or-equal-to 23"> <mml:semantics> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>23</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">5\leq p \leq 23</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime <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> and positive integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p minus"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>blocks with defect 0 in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a multiple of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for almost all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also establish that any given prime <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> divides the number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p minus"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p-</mml:annotation> </mml:semantics> </mml:math> </inline-formula>modularly irreducible representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">S_{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for almost all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
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>.
Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a …
Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup. Here we consider b(G) , the largest irreducible character degree of the group G . A simple application of Frobenius reciprocity shows that b(G) ≧ | G:A | for any abelian subgroup A of G . In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G) . If is G is nilpotent, a result of Isaacs and Passman (see [ 7 , Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G) 4 .
In this paper we continue our study of the relationship between the structure of a finite group G and the set of degrees of its irreducible complex characters.The following hypotheses …
In this paper we continue our study of the relationship between the structure of a finite group G and the set of degrees of its irreducible complex characters.The following hypotheses on the degrees are considered: (A) G has r.x. e for some prime p, i.e. all the degrees divide p e , (B) the degrees are linearly ordered by divisibility and all except 1 are divisible by exactly the same set of primes, (C) G has a.c.m, i.e., all the degrees except 1 are equal to some fixed m, (D) all the degrees except 1 are prime (not necessarily the same prime) and (E) all the degrees except 1 are divisible by p e > p but none is divisible by p e+1 .In each of these situations, group theoretic information is deduced from the character theoretic hypothesis and in several cases complete characterizations are obtained.
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 .
Introduction.The purpose of this paper is to obtain the character tables of the finite simple groups of Ree related to the Lie algebra G2 (presented in [16], [17]) from certain …
Introduction.The purpose of this paper is to obtain the character tables of the finite simple groups of Ree related to the Lie algebra G2 (presented in [16], [17]) from certain basic properties of these groups.In the process we shall derive a number of additional properties of the Ree groups.We incorporate the basic properties as conditions in the following definition:Definition.A finite group G will be said to be of Ree type if it satisfies the following five conditions: I.The 2-Sylow subgroups of G are elementary Abelian of order 8. II.G has no normal subgroup of index 2. III.For some element J of order 2 (an "involution") in G, the centralizer CG(J) of J in G is the direct product of <J> and L where L is isomorphic to the linear fractional group LF(2,q).Condition I implies that q = 4 + e (mod 8) where e = + 1.IV.If <R> denotes a cyclic subgroup of order (q + e)/2 in L, then the normalizer NC((R0}) of any subgroup <R0) ¥= <1> of <R> is contained in cG(J).V. Let J' be an involution of L and 5 an element of L of order (q -e)\\ which centralizes J'.Then an element of G of order 3 which normalizes <J, J'} does not centralize S.We call q the characteristic of G.The verification of these conditions is straightforward from the description of Ree's groups in [17].The existence of elements R, J', and S of IV and V is a consequence of the known structure of LF(2,q) which will be summarized in paragraph 1-1.Moreover, there is an involution J" of L commuting with J' for which J"SJ" = S~ , the centralizer of J' in Lis <J',J",5>.Thus the centralizer CG((J,J'}) is (J,J',J",Sy.Also, there is an element of order 3 in L normalizing <J',J"> but not centralizing it.
We complete the proof of the height conjecture for <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>-solvable groups, using the classification of finite simple groups.
We complete the proof of the height conjecture for <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>-solvable groups, using the classification of finite simple groups.
We classify the irreducible complex characters of the symplectic groups Sp 2n (q) and the orthogonal groups , Spin 2n+1(q) of degrees up to the bound D, where D = …
We classify the irreducible complex characters of the symplectic groups Sp 2n (q) and the orthogonal groups , Spin 2n+1(q) of degrees up to the bound D, where D = (q n − 1)q 4n−10/2 for symplectic groups, D = q 4n−8 for orthogonal groups in odd dimension, and D = q 4n−10 for orthogonal groups in even dimension.
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.
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)
Let Z be a normal subgroup of a finite group G, let λ ∈ Irr(Z) be an irreducible complex character of Z, and let p be a prime number.If p …
Let Z be a normal subgroup of a finite group G, let λ ∈ Irr(Z) be an irreducible complex character of Z, and let p be a prime number.If p does not divide the integers χ(1)/λ(1) for all χ ∈ Irr(G) lying over λ, then we prove that the Sylow p-subgroups of G/Z are abelian.This theorem, which generalizes the Gluck-Wolf Theorem to arbitrary finite groups, is one of the principal obstacles to proving the celebrated Brauer Height Zero Conjecture.
For a finite group H, let Irr(H) denote the set of irreducible characters of H, and define the 'zeta function' ζ H ( t ) = ∑ χ ∈ I …
For a finite group H, let Irr(H) denote the set of irreducible characters of H, and define the 'zeta function' ζ H ( t ) = ∑ χ ∈ I r r ( H ) χ ( 1 ) − t for real t > 0. We study the asymptotic behaviour of ζH(t) for finite simple groups H of Lie type, and also of a corresponding zeta function defined in terms of conjugacy classes. Applications are given to the study of random walks on simple groups, and on base sizes of primitive permutation groups. 2000 Mathematics Subject Classification 20C33, 20P05, 60B15, 20D06.
If a nontrivial nilpotent group $N$ acts faithfully and coprimely on a group $H$, it is shown that some element of $H$ has a small centralizer in $N$ and hence …
If a nontrivial nilpotent group $N$ acts faithfully and coprimely on a group $H$, it is shown that some element of $H$ has a small centralizer in $N$ and hence lies in a large orbit. Specifically, there exists $x \in H$ such that $|\mathbf {C}_{N}(x)| \le (|N|/p)^{1/p}$, where $p$ is the smallest prime divisor of $|N|$.
Let $G$ be a finite group and let $K$ be the conjugacy class of $x \in G$. If $K^2$ is a conjugacy class of $G$, then $[x,G]$ is solvable. If …
Let $G$ be a finite group and let $K$ be the conjugacy class of $x \in G$. If $K^2$ is a conjugacy class of $G$, then $[x,G]$ is solvable. If the order of $x$ is a power of prime, then $[x,G]$ has a normal $p$-complement. We also prove some related results on the solvability of certain normal subgroups when a non-trivial coset has certain properties.
In this paper it is shown that every finite set of powers of the prime <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> which contains <inline-formula …
In this paper it is shown that every finite set of powers of the prime <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> which contains <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript 0 Baseline equals 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{p^0} = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> occurs as the full set of degrees of the irreducible characters of some <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>-group.
and χ(l) = atθ(l).We have t = \G: T\ and ψ N = a Q θ, a 0 ^ a.Now χ is a constituent of ψ G and soWe have …
and χ(l) = atθ(l).We have t = \G: T\ and ψ N = a Q θ, a 0 ^ a.Now χ is a constituent of ψ G and soWe have equality throughout, so that χ(l) = ^G(l) and α = α 0 .ThusThe uniqueness of ψ also follows from a -α 0 -If e(G) = e and N < G, let 0 G irr( JNΓ) and Γ -^(0).Suppose that |G: T\ p = p r , where % denotes the p-part of the integer w.Let α/r be any irreducible constituent of θ τ , and let χ be an irreducible constituent of ψ G .Then by Frobenius reciprocity and Lemma 1.1, it follows that χ = ψ G and hence ^(l)j, ^ p e ~r.It does not follow, however, that e(T) ^ e -r.We wish to prove our results by induction in a manner similar to this and hence we define a quantity which "inducts" properly.DEFINITION 1.2.Let N < G and # e Irr(N).Suppose 0 is invariant in G. Then β(G, N, θ) = e is the largest integer such that ^e|(χ(l)/6'(l)) for some irreducible constituent χ of Θ G .Note that β(G, 1, 1) = e(G) and that if N^ H<\G, then e(H, N, θ) ^ e(G, N, θ) .The following is immediate.COROLLARY 1.3.Suppose e(G, N, θ) = e and N s M<\G.Let ψ be an irreducible constituent of Θ M and let p f = (ψ(l)/θ(l)) p .Set T = J*a(ψ) and p r = \G: T\ p .Then e(T, M, f) ^ e -f -r.P
This paper has two main results.Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for …
This paper has two main results.Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes.This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups.Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation.We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups.
We prove that every group of order n contains at least ɛ log n/(log log n)8 conjugacy classes for some fixed ɛ > 0. This essentially settles an old problem …
We prove that every group of order n contains at least ɛ log n/(log log n)8 conjugacy classes for some fixed ɛ > 0. This essentially settles an old problem of Brauer.
Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups …
Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups of symplectic type 397 5.4.^-groups, ^-solvability and F((&) 400 5.5.Groups of low order 404 5.6.2-groups, involutions and 2-length 409 5.7.Factorizations 423 5.8.Miscellaneous 426 6.A transitivity theorem 428 i 9 68]
All groups considered here are finite, unless otherwise specified.By Ch (G), we denote all complex characters of G; and by IRR (G), we denote the set of those $ e …
All groups considered here are finite, unless otherwise specified.By Ch (G), we denote all complex characters of G; and by IRR (G), we denote the set of those $ e Ch (G) that are irreducible.(On occasion, where it involves no loss of generality to t.he specific argument, we may say A s Ch (G) allowing the pos- sibility A 0).If a group A acts on G by automorphisms and if a defined by Za(ha) z(h) is an irreducible character of G whenever ;t is.If A is cyclic, the actions of A on IRR (G) and on the conjugacy classes of G are permutation isomorphic.Counterexamples exist for noncyclic A. We write IRRA (G) to denote the A-fixed irreducible characters of G. Now assume A acts on G by automorphisms and (I G I, IAI)= 1.Should A be solvable, G. Glauberman has defined a "natural" one-to-one correspon- dence between IRRa (G)and IRR (C)[5], if C C(A).When GI is odd, I.M. Isaacs has described a "natural" correspondence between IRRa (G)and IRR (C) [6].By "natural" we mean a map uniquely determined by the action of A on G and thus independent of choices made in an algorithm.The Odd-Order Theorem implies one of these correspondences occurs.One corollary of these correspondences is that A acts isomorphically on IRR (G) and the conjugacy classes of G (see comments preceding Theorem 5.5).Both correspondences exist precisely when GI is odd and A is solvable; and we show in this paper that the two are then identical.Let N<aG, zIRR(G), 0IRR(N), and T=It(0) (i.e., T= {9 e G 0 0}).We say Z s IRR (G[O) if [ZN, 0] :: 0. If has a unique irreducible constituent # s IRR (T 0).Also, Assume T G and Z s IRR (GIO).So XN =fO for somef Z.If f= 1; the constituents of 06 are precisely the characters fl;t for fl s IRR (G/N) and are distinct for distinct ft.This will occur whenever GIN is cyclic.If, on the other hand,f 2 G: N[ we say z or 0 is fully ramified with respect to GIN.This will occur if I6(0) G and either Z vanishes on G N or ;t is the unique constituent of 0.If K/L is an abelian chief factor of G and b IRR (K) is invariant in G; then b.IRR (L), b is fully ramified with respect to K/L, or tp. is the sum of ]K:L distinct irreducible characters of L. The results of these last few para- graphs are well known (see Chapter 6 of [7]) and will be used without reference.Section 2 basically deals with preliminaries.Sections 3 and 4 define and investigate the correspondences of Glauberman and Isaacs, respectively.Via
This book is based on a graduate course taught at the University of Paris. The authors aim to treat the basic theory of representations of finite groups of Lie type, …
This book is based on a graduate course taught at the University of Paris. The authors aim to treat the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it. They also discuss Deligne–Lusztig induction. This will be the first elementary treatment of this material in book form and will be welcomed by beginning graduate students in algebra.