Xiaoyou Chen

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Blocks of small defect in alternating groups and squares of Brauer character degrees

2017-07-18

Xiaoyou Chen , James P. Cossey , Mark L. Lewis +1 more

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

Super-Brauer characters and super-regular classes

2009-12-09

Xiaoyou Chen , Jiwen Zeng

Zeros of Brauer characters

2012-06-07

Huiqun Wang , Xiaoyou Chen , Jiwen Zeng

Finite $p$-groups with exactly two nonlinear non-faithful irreducible characters

2018-07-23

Ya-Li Li , Xiaoyou Chen , Huimin Li

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Brauer characters, degrees and subgroups

2022-04-25

Xiaoyou Chen , Gabriel Navarro

We prove a result on Brauer characters of finite groups, subgroups and degrees of characters, obtaining, as a corollary, a shorter proof of a generalization of a recent result of … We prove a result on Brauer characters of finite groups, subgroups and degrees of characters, obtaining, as a corollary, a shorter proof of a generalization of a recent result of G. Qian on element orders and character degrees.

ITÔ’S THEOREM AND MONOMIAL BRAUER CHARACTERS

2017-06-08

Xiaoyou Chen , Mark L. Lewis

Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$ -Brauer … Let $G$ be a finite solvable group and let $p$ be a prime. In this note, we prove that $p$ does not divide $\unicode[STIX]{x1D711}(1)$ for every irreducible monomial $p$ -Brauer character $\unicode[STIX]{x1D711}$ of $G$ if and only if $G$ has a normal Sylow $p$ -subgroup.

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Groups with only two nonlinear non-faithful irreducible characters

2017-12-14

Xiaoyou Chen , Mark L. Lewis

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SQUARES OF DEGREES OF BRAUER CHARACTERS AND MONOMIAL BRAUER CHARACTERS

2019-01-29

Xiaoyou Chen , Mark L. Lewis

Let $G$ be a finite group and let $p$ be a prime factor of $|G|$ . Suppose that $G$ is solvable and $P$ is a Sylow $p$ -subgroup of $G$ … Let $G$ be a finite group and let $p$ be a prime factor of $|G|$ . Suppose that $G$ is solvable and $P$ is a Sylow $p$ -subgroup of $G$ . In this note, we prove that $P{\vartriangleleft}G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all irreducible monomial $p$ -Brauer characters $\unicode[STIX]{x1D711}$ of $G$ .

Normal p-complements and monomial characters

2020-05-30

Xiaoyou Chen , Yong Yang

Groups Whose Nonlinear Irreduciblep-Brauer Characters are Real Valued

2015-10-19

Ya-Li Li , Jiwen Zeng , Xiaoyou Chen

Let p be a given prime. A finite group G whose all nonlinear irreducible p-Brauer characters are real valued is called a p-regular ℝ1-group. The aim of this article is … Let p be a given prime. A finite group G whose all nonlinear irreducible p-Brauer characters are real valued is called a p-regular ℝ1-group. The aim of this article is to show some results about the structures of p-regular ℝ1-groups. In particular, we will build the connections between p-regular ℝ1-groups and p-modular Frobenius groups. If these results apply to a finite group G with p∤|G|, we obtain similar results about finite groups whose nonlinear irreducible characters are real valued, and establish connections between these groups and Frobenius groups.

Linear p- Brauer characters and p-blocks

2017-04-03

Xiaoyou Chen , Jiwen Zeng

Let G be a finite group and p a prime divisor of | G |. Denote by IBr(G) and LBr(G) the sets of irreducible p-Brauer characters and linear p-Brauer characters … Let G be a finite group and p a prime divisor of | G |. Denote by IBr(G) and LBr(G) the sets of irreducible p-Brauer characters and linear p-Brauer characters of G, respectively. It is known that IBr(G) = LBr(G) if and only if P ∈ Sylp(G) is normal and G/P is abelian. We consider the local situation in this note. Let B be a p-block of G. We characterize the finite groups G with IBr(B) ⊆ LBr(G). And we get some results related to those of Holm, Willems and Isaacs, Smith.

Notes on conjugacy classes of finite groups

2018-11-17

Xiaoyou Chen , Huiqun Wang

We use elementary finite group theory to give direct and short proofs related to the results of conjugacy classes of finite groups. We use elementary finite group theory to give direct and short proofs related to the results of conjugacy classes of finite groups.

MONOLITHIC BRAUER CHARACTERS

2019-03-28

Xiaoyou Chen , Mark L. Lewis

Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$ . We say that a $p$ -Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We … Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$ . We say that a $p$ -Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$ . When $G$ is $p$ -solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$ -Brauer characters $\unicode[STIX]{x1D711}$ of $G$ .

Zeros of Monomial Brauer Characters

2019-01-17

Xiaoyou Chen , Gang Chen

Weak Mp-groups

2020-03-23

Xiaoyou Chen

Let G be a finite group and p be a prime. We prove in this note that if every irreducible monolithic p-Brauer character of G is monomial then G is … Let G be a finite group and p be a prime. We prove in this note that if every irreducible monolithic p-Brauer character of G is monomial then G is solvable.Communicated by J. Zhang

OD-Characterization of Some Simple Unitary Groups

2020-04-13

M. Akbari , Xiaoyou Chen , Alireza Moghaddamfar

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Solvable groups whose monomial, monolithic characters have prime power codegrees

2022-01-01

Xiaoyou Chen , Mark L. Lewis

In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$ is a $p$-power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi … In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$ is a $p$-power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi \in {\rm IBr} (G)$, then $P$ is normal in $G$, where $p$ is a prime and $P$ is a Sylow $p$-subgroup of $G$.

A Note on a Bound of Adan-Bante

2014-12-01

Xiaoyou Chen

Properties of Class Functions Associated to Subsections

2013-01-01

Xiaoyou Chen

Let G be a finite group,p a prime divisor of |G|,x a p-element of G and b be a p-block of CG(x).Then(x,b) is called a subsection of G provided bG … Let G be a finite group,p a prime divisor of |G|,x a p-element of G and b be a p-block of CG(x).Then(x,b) is called a subsection of G provided bG becomes a p-block of G.Subsections play an important role in modular representation theory.And let χ be an irreducible character of G;then χ(x,b) is a class function associated to(x,b).Some properties of this kind of class functions are given in this note and an example is also given to show how to obtain subsections.More concretely,let χ,ψ be irreducible characters of G,b,e be p-blocks of CG(x),and S be the complete set of representatives of G-classes of subsections of G.If b≠e,then [χ(x,b),ψ(x,e)]=0;if b=e,then the inner product of those two class functions has some connections with Cartan matrices.Moreover,χ=∑(x,b)∈Sχ(x,b).

Groups with one nonlinear irreducible Brauer character are solvable

2013-01-01

Xiaoyou Chen , Jiwen Zeng

Let G be a finite group and p be a prime divisor of |G|.Denote by IBr(G) and LBr(G) the sets of irreducible p-Brauer characters and linear p-Brauer characters of G, … Let G be a finite group and p be a prime divisor of |G|.Denote by IBr(G) and LBr(G) the sets of irreducible p-Brauer characters and linear p-Brauer characters of G, respectively.If all the irreducible p-Brauer characters of G are linear, that is, IBr(G) = LBr(G), then G is solvable.When G has only one nonlinear irreducible p-Brauer character, it is proved in this note that G is solvable.

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Groups with one or two super-Brauer character theories

2017-01-01

Xiaoyou Chen , Mark L. Lewis

A super-Brauer character theory of a group $G$ and a prime $p$ is a pair consisting of a partition of the irreducible $p$-Brauer characters and a partition of the $p$-regular … A super-Brauer character theory of a group $G$ and a prime $p$ is a pair consisting of a partition of the irreducible $p$-Brauer characters and a partition of the $p$-regular elements of $G$ that satisfy certain properties. We classify the groups and primes that have exactly one super-Brauer character theory. We discuss the groups with exactly two super-Brauer character theories.

Upper and lower bounds on Chillag table sums

2017-01-01

Xiaoyou Chen , Mark L. Lewis , Hung P. Tong‐Viet

Chillag has showed that there is a single generalization showing that the sums of ordinary character tables, Brauer character, and projective indecomposable characters are positive integers. We show that Chillag's … Chillag has showed that there is a single generalization showing that the sums of ordinary character tables, Brauer character, and projective indecomposable characters are positive integers. We show that Chillag's construction also applies to Isaacs' $π$-partial characters. We show that if an extra condition is assumed, then we can obtain upper and lower bounds on the Chillag's table sums. We will demonstrate that this condition holds for Brauer characters, $π$-partial characters, and projective indecomposable characters, and so we obtain upper and lower bounds for the table sums in those cases. We also obtain results regarding the sums of rows and columns in these tables.

Itô’s theorem and monomial brauer characters ii itô’s theorem and monomial brauer characters ii

2018-01-01

Xiaoyou Chen , Mark L. Lewis

An OD-Characterizable Class of Simple Groups

2018-06-06

M. Akbari , Xiaoyou Chen , Alireza Moghaddamfar

It is proved that finite nonabelian simple groups $S$ with $\max \pi(S)=37$ are uniquely determined by their order and degree pattern in the class of all finite groups. It is proved that finite nonabelian simple groups $S$ with $\max \pi(S)=37$ are uniquely determined by their order and degree pattern in the class of all finite groups.

Some properties of various graphs associated with finite groups

2018-01-01

Xiaoyou Chen , Alireza Moghaddamfar , M. Zohourattar

In this paper we have investigated some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it has been shown … In this paper we have investigated some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it has been shown that the simple group $L_2(7)$ can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, we have obtained an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.

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Itô's theorem and monomial Brauer characters

2017-01-01

Xiaoyou Chen , Mark L. Lewis

Let $G$ be a finite solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $\varphi(1)$ for every irreducible monomial $p$-Brauer character … Let $G$ be a finite solvable group, and let $p$ be a prime. In this note, we prove that $p$ does not divide $\varphi(1)$ for every irreducible monomial $p$-Brauer character $\varphi$ of $G$ if and only if $G$ has a normal Sylow $p$-subgroup.

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DEGREES OF BRAUER CHARACTERS AND NORMAL SYLOW -SUBGROUPS

2020-01-08

Xiaoyou Chen , Mark L. Lewis

Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$ -subgroup of $G$ . We prove that $P$ is normal in $G$ if and only if … Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$ -subgroup of $G$ . We prove that $P$ is normal in $G$ if and only if $\unicode[STIX]{x1D711}(1)_{p}^{2}$ divides $|G:\ker (\unicode[STIX]{x1D711})|_{p}$ for all monomial monolithic irreducible $p$ -Brauer characters $\unicode[STIX]{x1D711}$ of $G$ .

A note on monolithic Brauer characters

2021-01-01

Xiaoyou Chen , Ni Du

A note on $$\pi $$-partial characters of $$\pi $$-separable groups

2021-02-11

Xiaoyou Chen , Yong Yang

Normal Sylow subgroups and monomial Brauer characters

2021-08-28

Xiaoyou Chen , Лонг Миао

Notes on normal p-complements and monomial characters

2021-10-27

Xiaoyou Chen , Yong Yang

Let G be a finite group and Irr(G) be the set of irreducible (complex) characters of G. Let χ ∈ Irr(G) and write cod(χ) = |G : ker χ|/χ(1) as … Let G be a finite group and Irr(G) be the set of irreducible (complex) characters of G. Let χ ∈ Irr(G) and write cod(χ) = |G : ker χ|/χ(1) as the codegree of χ, where ker χ is the kernel of χ. Denote by the intersection of the kernels of all the monolithic, monomial irreducible characters of a p-solvable group G with codegrees divisible by a prime p. We prove in this note that has a normal p-complement, and then the theorem in [2] is generalized. Also, we prove that about the theorem in [2] more is true.

Upper and Lower Bounds of Table Sums

2021-11-08

Xiaoyou Chen , Mark L. Lewis , Hung P. Tong‐Viet

For a group [Formula: see text], we produce upper and lower bounds for the sum of the entries of the Brauer character table of [Formula: see text] and the projective … For a group [Formula: see text], we produce upper and lower bounds for the sum of the entries of the Brauer character table of [Formula: see text] and the projective indecomposable character table of [Formula: see text]. When [Formula: see text] is a [Formula: see text]-separable group, we show that the sum of the entries in the table of Isaacs' partial characters is a real number, and we obtain upper and lower bounds for this sum.

An OD-Characterizable Class of Simple Groups

2018-01-01

Majid Akbari , Xiaoyou Chen , Alireza Moghaddamfar

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Solvable groups whose monomial, monolithic characters have prime power codegrees

2023-12-01

Xiaoyou Chen , Mark L. Lewis

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On the odd order subgroups of solvable linear groups

2024-03-31

Fang Bian , Xiaoyou Chen , Yong Yang

In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the … In this paper, we first strengthen a few results about the order of solvable linear groups of odd order. We then use these results to bound the product of the orders of odd-order composition factors in a composition series of an arbitrary finite linear group.

Solving Multi-Point Boundary Value Differential Equations with a Novel Semi-Analytical Algorithm

2024-12-11

Samad Kheybari , M.T. Darvishi , Farzaneh Alizadeh +3 more

A generalization on Bessenrodt’s corollary

2024-12-11

Xiaoyou Chen , Ni Du

Let G be a finite group, p be a prime and let IBr(G) be the set of irreducible (p−) Brauer characters of G. If every Brauer character φ∈IBr(G) is monomial, … Let G be a finite group, p be a prime and let IBr(G) be the set of irreducible (p−) Brauer characters of G. If every Brauer character φ∈IBr(G) is monomial, then G is said to be an Mp-group. Bessenrodt proved that normal Hall subgroups of Mp-groups are Mp-groups. In this article, we generalize this result.

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

2025-05-09

Xiaoyou Chen , Mark L. Lewis

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

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

2025-05-09

Xiaoyou Chen , Mark L. Lewis

Solving Multi-Point Boundary Value Differential Equations with a Novel Semi-Analytical Algorithm

2024-12-11

Samad Kheybari , M.T. Darvishi , Farzaneh Alizadeh +3 more

A generalization on Bessenrodt’s corollary

2024-12-11

Xiaoyou Chen , Ni Du

On the odd order subgroups of solvable linear groups

2024-03-31

Fang Bian , Xiaoyou Chen , Yong Yang

Solvable groups whose monomial, monolithic characters have prime power codegrees

2023-12-01

Xiaoyou Chen , Mark L. Lewis

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Brauer characters, degrees and subgroups

2022-04-25

Xiaoyou Chen , Gabriel Navarro

Solvable groups whose monomial, monolithic characters have prime power codegrees

2022-01-01

Xiaoyou Chen , Mark L. Lewis

Upper and Lower Bounds of Table Sums

2021-11-08

Xiaoyou Chen , Mark L. Lewis , Hung P. Tong‐Viet

Notes on normal p-complements and monomial characters

2021-10-27

Xiaoyou Chen , Yong Yang

Normal Sylow subgroups and monomial Brauer characters

2021-08-28

Xiaoyou Chen , Лонг Миао

A note on $$\pi $$-partial characters of $$\pi $$-separable groups

2021-02-11

Xiaoyou Chen , Yong Yang

A note on monolithic Brauer characters

2021-01-01

Xiaoyou Chen , Ni Du

Normal p-complements and monomial characters

2020-05-30

Xiaoyou Chen , Yong Yang

OD-Characterization of Some Simple Unitary Groups

2020-04-13

M. Akbari , Xiaoyou Chen , Alireza Moghaddamfar

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Weak Mp-groups

2020-03-23

Xiaoyou Chen

DEGREES OF BRAUER CHARACTERS AND NORMAL SYLOW -SUBGROUPS

2020-01-08

Xiaoyou Chen , Mark L. Lewis

MONOLITHIC BRAUER CHARACTERS

2019-03-28

Xiaoyou Chen , Mark L. Lewis

SQUARES OF DEGREES OF BRAUER CHARACTERS AND MONOMIAL BRAUER CHARACTERS

2019-01-29

Xiaoyou Chen , Mark L. Lewis

Zeros of Monomial Brauer Characters

2019-01-17

Xiaoyou Chen , Gang Chen

Notes on conjugacy classes of finite groups

2018-11-17

Xiaoyou Chen , Huiqun Wang

Finite $p$-groups with exactly two nonlinear non-faithful irreducible characters

2018-07-23

Ya-Li Li , Xiaoyou Chen , Huimin Li

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An OD-Characterizable Class of Simple Groups

2018-06-06

M. Akbari , Xiaoyou Chen , Alireza Moghaddamfar

Itô’s theorem and monomial brauer characters ii itô’s theorem and monomial brauer characters ii

2018-01-01

Xiaoyou Chen , Mark L. Lewis

Some properties of various graphs associated with finite groups

2018-01-01

Xiaoyou Chen , Alireza Moghaddamfar , M. Zohourattar

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An OD-Characterizable Class of Simple Groups

2018-01-01

Majid Akbari , Xiaoyou Chen , Alireza Moghaddamfar

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Groups with only two nonlinear non-faithful irreducible characters

2017-12-14

Xiaoyou Chen , Mark L. Lewis

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Blocks of small defect in alternating groups and squares of Brauer character degrees

2017-07-18

Xiaoyou Chen , James P. Cossey , Mark L. Lewis +1 more

ITÔ’S THEOREM AND MONOMIAL BRAUER CHARACTERS

2017-06-08

Xiaoyou Chen , Mark L. Lewis

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Linear p- Brauer characters and p-blocks

2017-04-03

Xiaoyou Chen , Jiwen Zeng

Groups with one or two super-Brauer character theories

2017-01-01

Xiaoyou Chen , Mark L. Lewis

Upper and lower bounds on Chillag table sums

2017-01-01

Xiaoyou Chen , Mark L. Lewis , Hung P. Tong‐Viet

Itô's theorem and monomial Brauer characters

2017-01-01

Xiaoyou Chen , Mark L. Lewis

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Groups Whose Nonlinear Irreduciblep-Brauer Characters are Real Valued

2015-10-19

Ya-Li Li , Jiwen Zeng , Xiaoyou Chen

A Note on a Bound of Adan-Bante

2014-12-01

Xiaoyou Chen

Properties of Class Functions Associated to Subsections

2013-01-01

Xiaoyou Chen

Groups with one nonlinear irreducible Brauer character are solvable

2013-01-01

Xiaoyou Chen , Jiwen Zeng

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Zeros of Brauer characters

2012-06-07

Huiqun Wang , Xiaoyou Chen , Jiwen Zeng

Super-Brauer characters and super-regular classes

2009-12-09

Xiaoyou Chen , Jiwen Zeng

Coauthor	Papers Together
Mark L. Lewis	14
Jiwen Zeng	5
Yong Yang	4
Alireza Moghaddamfar	4
Hung P. Tong‐Viet	3
Ni Du	2
Ya-Li Li	2
M. Akbari	2
Farzaneh Alizadeh	1
Gabriel Navarro	1
K. Hosseini	1
Samad Kheybari	1
Majid Akbari	1
Evren Hınçal	1
Huiqun Wang	1
M.T. Darvishi	1
Huimin Li	1
Fang Bian	1
Huiqun Wang	1
Лонг Миао	1
M. Zohourattar	1
Gang Chen	1
James P. Cossey	1

Character Theory of Finite Groups

1976-01-01

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Character theory of finite groups

1999-07-01

I. M. Isaacs

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

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Finite groups and degrees of irreducible monomial characters

2015-04-27

Linna Pang , Jiakuan Lu

Let [Formula: see text] be a finite solvable group, let Irr[Formula: see text] be the set of all irreducible monomial characters of [Formula: see text] and let [Formula: see text] … Let [Formula: see text] be a finite solvable group, let Irr[Formula: see text] be the set of all irreducible monomial characters of [Formula: see text] and let [Formula: see text] be a prime. We prove that if [Formula: see text] for every nonlinear [Formula: see text][Formula: see text][Formula: see text], then [Formula: see text] has a normal [Formula: see text]-complement, and if [Formula: see text] is relatively prime to [Formula: see text] for every [Formula: see text], then [Formula: see text] has a normal Sylow [Formula: see text]-subgroup.

Group Characters and Normal Hall Subgroups

1962-12-01

P. X. Gallagher

1. Introduction Let G be a finite group and let ψ be an (ordinary) irreducible character of a normal subgroup N . If ψ extends to a character of G … 1. Introduction Let G be a finite group and let ψ be an (ordinary) irreducible character of a normal subgroup N . If ψ extends to a character of G then ψ is invariant under G , but the converse is false. In section 3 it is shown that if ψ extends coherently to the intermediate groups H for which H/N is elementary, then ψ extends to G . If N is a Hall subgroup, then in order for ψ to extend to G it is sufficient that ψ be invariant under G . This leads to a construction of the characters of G from the characters of N and the characters of the subgroups of G/N in this case.

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Large orbits in actions of nilpotent groups

1999-01-01

I. M. Isaacs

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 $|N|$.

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Zeros of Brauer characters

2012-06-07

Huiqun Wang , Xiaoyou Chen , Jiwen Zeng

Module correspondence in finite groups

1981-02-01

Tetsuro Okuyama

In section 5, we try to give some refinement of Brauer's induction theorem in terms of p -blocks.Dress's induction theorem is also considered there.Let G be a finite group and … In section 5, we try to give some refinement of Brauer's induction theorem in terms of p -blocks.Dress's induction theorem is also considered there.Let G be a finite group and p a fixed rational prime number.In this paper we use the following notations and terminologies and for other notations and terminologies we shall refer to books of Gorenstein [16], Dornhoff [8] and Feit [11].R ; a complete discrete valuation ring with maximal ideal (\pi)\ni p .K ; the quotient field of R. F;=R/(\pi) , the residue field of R which has characteristic; the group algebras of G over R, K and F respec- tively.We assume that fields K and F are both splitting fields for all groups considered in this paper.All modules considered are right unital and finitely generated.V^{N} ; the induced module for an F[H] -module V to N where H and N are subgroups of G with H\subseteqq N .V_{1H} ; the restriction of an F[N] -module V to H where H and N are subgroups of G with H\subseteqq N .V^{x} ; the conjugate module of an F[H] -module V by an element x in G which is an F[H^{x}] -module.If H is a normal subgroup of G, we define the inertia subgroup of V, denoted by I_{G}(V) , to be the set \{x\in G|V^{x}\cong V\} .V|W ; for F[G] modules V and W, V is isomorphic to a direct summand of W.In the above we use the same notations for R[T] -modules too for a subgroup T of G. L_{0}(G) ; an F[G] -module of dimension 1 on which G acts trivially.For F[G] modules V and W and a subgroup H of G define ( V, W)_{H}=Hom_{F[H]} ( V, W) .The relative norm map T_{H,G} ; ( V, W)_{H}arrow(V, W)_{G} is defined as follows.T_{H,G}( \lambda)=\sum_{x}\lambda^{x} for \lambda\in(V, W)_{H} , where x ranges over a set of representatives of right cosets of H in G and \lambda^{x} is the map varrow(\lambda(vx^{-1}))x .If \mathfrak{H}_{\vee} is a set of subgroups of G, we define (V, W)_{G}^{\mathfrak{H}}=(V, W)_{G}/(V, W)_{\mathfrak{H},G} where (V, W)_{\mathfrak{H},G}= \sum_{H\in \mathfrak{H}} Im T_{H,G} .Now we shall describe the Green correspondence which is frequently used in the paper.We refer to a book of Feit [11].Let V be an indecomposable F[G] -module.Then there exists a psubgroup D which is a minimal subgroup of G such that V is F[D] -projec- tive.A group D is uniquely determined up to conjugation by an element

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A character theoretic condition characterizing nilpotent groups

1999-01-01

Stephen M. Gagola , Mark L. Lewis

Characters of Π-separable groups

1984-01-01

I. M. Isaacs

Blocks of small defect in alternating groups and squares of Brauer character degrees

2017-07-18

Xiaoyou Chen , James P. Cossey , Mark L. Lewis +1 more

Brauer characters and normal Sylow p-subgroups

2018-02-14

Hung P. Tong‐Viet

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Finite simple groups with narrow prime spectrum

2008-01-01

Andrei V. Zavarnitsine

We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor … We find the nonabelian finite simple groups with order prime divisors not exceeding 1000. More generally, we determine the sets of nonabelian finite simple groups whose maximal order prime divisor is a fixed prime less than 1000. Our results are based on calculations in the computer algebra system GAP.

Super-Brauer characters and super-regular classes

2009-12-09

Xiaoyou Chen , Jiwen Zeng

SQUARES OF DEGREES OF BRAUER CHARACTERS AND MONOMIAL BRAUER CHARACTERS

2019-01-29

Xiaoyou Chen , Mark L. Lewis

Finite groups having only one irreducible representation of degree greater than one

1968-04-01

Gary M. Seitz

In this paper all finite groups having exactly one irreducible K-representation of degree greater than one are determined, where K is an algebraically closed field of characteristic zero. The quaternion … In this paper all finite groups having exactly one irreducible K-representation of degree greater than one are determined, where K is an algebraically closed field of characteristic zero. The quaternion group of order eight and the dihedral group of order eight are nilpotent groups with this property, while the symmetric group on three letters and the alternating group on four letters are solvable although not nilpotent examples. The theorem below will show how typical the above examples really are. In the following all groups are finite. If G is a group, let GI denote the derived group of G and Z(G) the center of G. Let K be an algebraically closed field of characteristic zero.

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Element orders and character codegrees

2011-10-17

I. M. Isaacs

The co-degrees of irreducible characters

1991-06-01

David Chillag , Avinoam Mann , Olaf Manz

MONOLITHIC BRAUER CHARACTERS

2019-03-28

Xiaoyou Chen , Mark L. Lewis

Characters of Solvable Groups

2018-05-23

I. M. Isaacs

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A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

2009-06-14

Carlos A. M. André , Ana Margarida Neto

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Finite Groups Having Only One Irreducible Representation of Degree Greater than One

1968-04-01

Gary M. Seitz

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A Characterization of PSU3(q) for q ? 5

2003-01-01

Ali Iranmanesh , Behrooz Khosravi , Seyed Hassan Alavi

ITÔ’S THEOREM AND MONOMIAL BRAUER CHARACTERS

2017-06-08

Xiaoyou Chen , Mark L. Lewis

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On the Prime Graph of a Finite Simple Group An Application of the Method of Feit-Thompson-Bender-Glauberman

2018-12-29

Michio Suzuki

Theorem A. Let ~ be a connected component of the prime gmph r(G) of a finite group G, and let ro be the set of primes in~-Assume that ~ =f.r( … Theorem A. Let ~ be a connected component of the prime gmph r(G) of a finite group G, and let ro be the set of primes in~-Assume that ~ =f.r( G) and 2 ¢: ro.Then, ~ is a clique.Usually, we identify ~ with ro and abuse the terms, saying ro is a connected component of the graph r( G).Theorem A has not been stated

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PRIME GRAPH COMPONENTS OF FINITE SIMPLE GROUPS

1990-02-28

А. С. Кондратьев

Let G be a finite group and π(G) the set of prime factors of its order. The prime graph of G is the graph with vertex-set π(G), two vertices p … Let G be a finite group and π(G) the set of prime factors of its order. The prime graph of G is the graph with vertex-set π(G), two vertices p and q being joined by an edge whenever G contains an element of order pq. This article contains an explicit description of the primes in each of the connected components of the prime graphs of the finite simple groups of Lie type of even characteristic. This solves question 9.16 of the Kourovka Notebook. Bibliography: 15 titles.

Codegrees and nilpotence class of p-groups

2015-12-17

Ni Du , Mark L. Lewis

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

A Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

2005-09-01

A. R. Moghaddamfar , A. R. Zokayi , M. R. Darafsheh

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices … If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p 1 <p 2 <⋯<p k and natural numbers α i . For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p 1 ), deg (p 2 ),…, deg (p k )). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups A p with p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={2 9 .3 9 .5.7, 13}, then G≅S 6 (3) or O 7 (3), and finally, we show that if G is a finite group such that |G|=2 9 .3 9 .5.7.13 and D(G)=(3,2,2,1,0), then G≅S 6 (3) or O 7 (3).

Supercharacter formulas for pattern groups

2009-04-01

Persi Diaconis , Nathaniel Thiem

C. Andre and N. Yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group $U_n(\mathbb … C. Andre and N. Yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group $U_n(\mathbb {F}_q)$. In this theory superclasses are certain unions of conjugacy classes, and supercharacters are a set of characters which are constant on superclasses. This paper gives a character formula for a supercharacter evaluated at a superclass for pattern groups and more generally for algebra groups.

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The analog of Huppert's conjecture on character codegrees

2016-12-30

Yong Yang , Guohua Qian

On a theorem of Gagola and Lewis

2016-08-17

Jiakuan Lu

Gagola and Lewis proved that a finite group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] … Gagola and Lewis proved that a finite group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible characters [Formula: see text] of [Formula: see text]. In this paper, we prove that a finite soluble group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible monomial characters [Formula: see text] of [Formula: see text].

Groups with the same order and degree pattern

2011-12-05

Roya Kogani-Moghaddam , A. R. Moghaddamfar

Supercharacters of the Sylowp-subgroups of the finite symplectic and orthogonal groups

2009-01-01

Carlos A. M. André , Ana Margarida Neto

We define and study supercharacters of the classical finite unipotent groups of types B n (q), C n (q) and D n (q).We show that the results we proved in … We define and study supercharacters of the classical finite unipotent groups of types B n (q), C n (q) and D n (q).We show that the results we proved in 2006 remain valid over any finite field of odd characteristic.In particular, we show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters.In addition, we prove that the unitary vector space spanned by all the supercharacters is closed under multiplication, and establish a formula for the supercharacter values.As a consequence, we obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters.Finally, we give a combinatorial description of all the irreducible characters of maximum degree in terms of the root system, by showing how they can be obtained as constituents of particular supercharacters.

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Zeros of Primitive Characters in Solvable Groups

1999-11-01

Gabriel Navarro

Co-degrees of irreducible characters in finite groups

2006-12-13

Guohua Qian , Yanming Wang , Huaquan Wei

OD-Characterization of the Projective Special Linear Groups L2(q)

2012-07-05

Liangcai Zhang , Wujie Shi

Let L 2 (q) be the projective special linear group, where q is a prime power. In the present paper, we prove that L 2 (q) is OD-characterizable by using … Let L 2 (q) be the projective special linear group, where q is a prime power. In the present paper, we prove that L 2 (q) is OD-characterizable by using the classification of finite simple groups. A new method is introduced in order to deal with the subtle changes of the prime graph of a group in the discussion of its OD-characterization. This not only generalizes a result of Moghaddamfar, Zokayi and Darafsheh, but also gives a positive answer to a conjecture put forward by Shi.

Classification of solvable groups possessing a unique nonlinear non-faithful irreducible character

2013-10-29

Amin Saeidi

Abstract In this note, we study finite groups possessing exactly one nonlinear non-faithful irreducible character. Our main result is to classify solvable groups that satisfy this property. Also, we give … Abstract In this note, we study finite groups possessing exactly one nonlinear non-faithful irreducible character. Our main result is to classify solvable groups that satisfy this property. Also, we give examples to show that these groups need not to be solvable in general.

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Supercharacters and superclasses for algebra groups

2007-11-20

Persi Diaconis , I. M. Isaacs

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

Finite groups and elements where characters vanish

2010-04-12

Gang Chen

Prime graph components of finite groups

1981-04-01

Julian Williams

Characters of Finite Groups. Part 2

1998-09-29

Ya. Berkovich , Emmanuel M. Zhmud'

Normal p-complements and irreducible characters

1970-02-01

John G. Thompson

Semisimple Commutative Algebras with Positive Bases

1998-12-01

David Chillag

Orbit sizes and character degrees, II

1999-11-22

Thomas Michael Keller

A Quantitative Characterization of Some Finite Simple Groups Through Order and Degree Pattern

2015-02-15

A. R. Moghaddamfar , S. Rahbariyan

Let be a finite group with , where are prime numbers and are natural numbers. The prime graph of is a simple graph whose vertex set is and two distinct … Let be a finite group with , where are prime numbers and are natural numbers. The prime graph of is a simple graph whose vertex set is and two distinct primes and are joined by an edge if and only if has an element of order . The degree of a vertex is the number of edges incident on , and the -tuple is called the degree pattern of . We say that the problem of OD-characterization is solved for a finite group if we determine the number of pairwise non-isomorphic finite groups with the same order and degree pattern as . The purpose of this paper is twofold. First, it completely solves the OD-characterization problem for every finite non-Abelian simple groups their orders having prime divisors at most 17. Second, it provides a list of finite (simple) groups for which the problem of OD-characterization have been already solved.

Recognizing Finite Groups Through Order and Degree Pattern

2008-09-01

A. R. Moghaddamfar , A. R. Zokayi

The degree pattern of a finite group G is introduced in [10] and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: … The degree pattern of a finite group G is introduced in [10] and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: all sporadic simple groups, alternating groups A p (p ≥ 5 is a twin prime) and some simple groups of Lie type. In this paper, we continue this investigation. In particular, we show that the automorphism groups of sporadic simple groups (except Aut (J 2 ) and Aut (M c L)), all simple C 22 -groups, the alternating groups A p , A p+1 , A p+2 and the symmetric groups S p , S p+1 , where p is a prime, are also uniquely determined by their degree patterns and orders.

Defect zero blocks for finite simple groups

1996-01-01

Andrew Granville , Ken Ono

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>.

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Some Studies on Group Characters

1951-02-01

Noboru Ιτο

The theory of group characters was originated by G. Frobenius and has been studied by many authors including, above all, I. Schur and W. Burnside. As to the modular theory … The theory of group characters was originated by G. Frobenius and has been studied by many authors including, above all, I. Schur and W. Burnside. As to the modular theory we owe it to recent works by R. Brauer and his collaborators, including T. Nakayama and C. Nesbitt, which clarified also the connections between the structure theory and the representation theory.

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