On p-parts of character degrees and conjugacy class sizes of finite groups

Abstract

Let G be a finite group and Irr(G) the set of irreducible complex characters of G. Let ep(G) be the largest integer such that p ep(G) divides χ(1) for some χ ∈ Irr(G). We show that |G : F(G)|p ≤ p kep(G) for a constant k. This settles a conjecture of A. Moretó [17, Conjecture 4].We also study the related problems of the p-parts of conjugacy class sizes of finite groups.

Kernels of p'-degree irreducible characters

“…Corollary C shows that, in fact, |O p ′ ,p (Sol(G))| p is the only factor of |G| p that cannot be bounded in terms of the largest p-part of the character codegrees. The following consequence of Corollary C has a very similar character degree analog (see Corollary B of [12] and Theorem 3.12 of [16]). We need a lemma first.…”

“…It was conjectured in [10] that |G : O p (G)| p is bounded in terms of the largest p-part of the character degrees of G. This conjecture was proved for solvable groups in Corollary B of [12] and for arbitrary groups in Theorem A of [16]. The natural question, therefore, is whether such a bound exists if we replace character degrees by character codegrees.…”

On p-parts of Brauer character degrees and p-regular conjugacy class sizes

“…We now state the orbit theorem for p-solvable groups. This result has been proved in [34] but the proof there has some glitch; we take the opportunity to provide a corrected proof here.…”

“…For p > 2, Lewis, Navarro, Tiep and Tong-Viet [20] also studied the case when e p (G) = 1 for arbitrary finite groups. The conjecture of Moretó was recently settled by Qian and the second author in [34].…”

On p-parts of character degrees

“…Moretó [12,Conjecture 4] conjectured that log p b(P ) is bounded by some function of e p (G). The conjecture was first proved for solvable groups by Moretó and Wolf using an orbit theorem of solvable linear groups in [13], and later settled by the authors for arbitrary finite groups in [21]. In fact, we showed that log p |G/O p (G)| p is bounded by a linear function of e p (G).…”