Zeros of irreducible characters in factorised groups

Abstract

An element g of a finite group G is said to be vanishing in G if there exists an irreducible character χ of G such that χ(g) = 0; in this case, g is also called a zero of G. The aim of this paper is to obtain structural properties of a factorised group G = AB when we impose some conditions on prime power order elements g ∈ A ∪ B which are (non-)vanishing in G.Keywords Finite groups · Products of groups · Irreducible characters · Conjugacy classes · Vanishing elements 2010 MSC 20D40 · 20C15 · 20E45 Note that if we adopt the bar convention for the quotients over K, the above condition means that A G B G = 1, where H X denotes the core in a group X of a subgroup H. This illustrates the given name for such factorisations.Remark 2.2. Let state some immediate facts:1. If either 1 = G = A or 1 = G = B, then G = AB is always a core-factorisation.2. If G = AB is a core-factorisation of a simple group G, then either G = A or G = B.

On products of groups and indices not divisible by a given prime

“…Other papers consider mutually permutable products (see [2,7,11]). Recent work done by some of the authors ( [13,14]) extends previous developements by considering some special type of factorisations, the so-called core-factorisations. Only [12] treats prime power indices without considering any additional restriction on the factors.…”

Products of Groups and Class Sizes of $$\pi $$-Elements

“…A detailed account on this topic can be found in the book [3]. Throughout this paper, we deal with products of groups that possess a especial chief series, the so-called core-factorisations, introduced in [11] (see also Definition 2.1).…”

“…We also provided a useful characterisation of core-factorisations via quotients (compare with [11,Lemma 2]). Lemma 2.4.…”

“…Moreover, as noted in [11,Example 2], if N is an arbitrary prefactorised normal subgroup of a core-factorisation G = AB, then N = (N ∩ A)(N ∩ B) may not be a core-factorisation. Nevertheless, this condition behaves well for quotients of G. Next we show some series constructions for core-factorisations somehow similar to the lower/upper π-series of a π-separable group.…”

“…On the other hand, a current activity shows up that, in a factorised group, the sizes of the conjugacy classes in the group of the elements in the factors have a strong impact on the structure of the whole group (see, for instance, [2,10,11,14]). Our main purpose here is to study the π-structure of groups with a core-factorisation when the class lengths in the group of the π-elements in the factors are either π-numbers or π ′ -numbers, for a set of primes π.…”