Groups with few $p'$-character degrees

2019

Abstract We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group G contains only two distinct values not divisible by a given prime number p > 3 , then O p p ′ p p ′ ( G ) = 1 . This is done by using the classification of finite simple groups.

Restricting irreducible characters to Sylow 𝑝-subgroups

2018

We restrict irreducible characters of finite groups of degree divisible by p p to their Sylow p p -subgroups and study the number of linear constituents.

Restriction of odd degree characters and natural correspondences

2016

Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a natural correspondence of characters, both for $GL_n(q)$ and $SL_n(q)$. A similar result is established for certain finite groups with self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of $S_n$ and those of $M$, where $M$ is any maximal subgroup of $S_n$ of odd index; as well as between the odd-degree irreducible characters of $G = GL_n(q)$ or $GU_n(q)$ with $q$ odd and those of $N_{G}…

Irreducible characters of $3'$-degree of finite symmetric, general linear and unitary groups

2018

Abstract Let G be a finite symmetric, general linear, or general unitary group defined over a field of characteristic coprime to 3. We construct a canonical correspondence between irreducible characters of degree coprime to 3 of G and those of N G ( P ) , where P is a Sylow 3-subgroup of G . Since our bijections commute with the action of the absolute Galois group over the rationals, we conclude that fields of values of character correspondents are the same.

Groups with few p′-character degrees

2020