Groups with few $p'$-character degrees

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

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

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}…

Groups with few p′-character degrees

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

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.