Primes dividing the degrees of the real characters

Abstract

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito-Michler Theorem asserts that if a prime p does not divide the degree of any χ ∈ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a realvalued version of this theorem, where instead of Irr(G) we only consider the subset Irr rv (G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr rv (G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.

Finite groups with real-valued irreducible characters of prime degree

“…We shall also make use of the following theorems, which relate the arithmetical structure of cd rv (G) and the group structure of G. The first one is Theorem 4.2 in [4]. A 2-group T such that cd rv (T ) = {1} will be called a 2-group of Chillag-Mann type, as this class of groups was studied in [2].…”

“…It may be worth mentioning that our proof of Theorem A depends indirectly on the Classification of Finite Simple Groups, as it involves an application of Theorem 4.2 in [4]. Instead, with a direct use of the Classification, we can prove that a finite group whose real-valued irreducible characters have all squarefree degree is either solvable or an extension of a solvable group by the alternating group A 7 (Theorem 3.1).…”

Coprime action and arithmetical conditions on invariant conjugacy classes

“…This has been generalized in several directions by restricting the hypothesis to certain proper subsets of classes. For example, Camina [3] extended it for conjugacy classes of elements of p -order: All class sizes of p -elements of G are not divisible by p if and only if the Sylow p-subgroup is a direct factor of G. More recently, Dolfi et al [9] have shown that all the real conjugacy classes of G have odd size if and only if G has a normal Sylow 2-subgroup, and consequently, G is solvable too. It is difficult to summarize here the many results on class sizes appeared in the literature concerning solvability, so we refer the reader to the excellent survey [5].…”

Irreducible characters of even degree and normal Sylow 2-subgroups

“…One primary direction is to weaken the condition that all irreducible characters of G have degree coprime to p, and assume instead only that a subset of characters with a specified field of values has this property, see [4,19] for real-valued characters and [25] for p-rational characters.…”