Itô’s Theorem and Monomial Brauer characters

Abstract

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.

Itô’s Theorem and Monomial Brauer Characters Ii

“…Note that if p does not divide the degree of every monomial irreducible Brauer character, then IBr m,p (G) = ∅ and so M = G. By Theorem 1, G is p-closed. Conversely, as we noted in[1], if G is p-closed, then p does not divide the degree of any irreducible Brauer character.…”

“…In [5], Pang and Lu proved that when G is solvable and there is a prime p so that p does not divide the degree of any monomial irreducible character, then G is p-closed. In our paper [1], we mistakenly stated that they also proved the converse. When G is a nonabelian p-group, it is p-closed and has at least one monomial irreducible character whose degree is divisible by p. Thus, not only did Pang and Lu not prove the converse; in fact, the converse is not true.…”

“…For Brauer characters of p-solvable groups, Itô recovered the normality of Sylow subgroups (see [3,Theorem 13.1(b) and (c)]). We generalised this result in [1] for solvable groups by only using the monomial p-Brauer characters. Following the idea of Pang and Lu, we now let IBr m,p (G) be the set of irreducible monomial p-Brauer characters of G whose degrees are divisible by p and we define M to be the intersection Proof.…”

Monolithic Brauer Characters

“…The Brauer character version of the Itô-Michler theorem says that P is a normal Sylow p-subgroup of a group G if and only if p ϕ(1) for every Brauer character ϕ ∈ IBr(G) (see [14,Theorem 3.1]). With the hypothesis that G is solvable, we proved in [3] In [5], Gagola and the second author characterised nilpotent groups through a character-theoretic condition. In fact, they proved that a group G is nilpotent if and only if χ(1) 2 divides |G : ker χ| for all χ ∈ Irr(G).…”