Irreducible products of characters.
type: Item
instance of: scholarly article
title: Irreducible products of characters.
zbMATH Open document ID: 0947.20007
author: I. M. Isaacs
published in: Journal of Algebra
publication date: 2000-11-05T00:00:00Z
review text: Let \(G\) be a finite group, \(\alpha,\beta\in\text{Irr}(G)\). In this paper the author asks when it can happen that the product \(\alpha\beta\in\text{Irr}(G)\). He considers the most interesting case where \(\alpha\) and \(\beta\) are faithful. ``We have been unable to find a noncyclic solvable example (with irreducible \(\alpha\beta\)), however, and we suspect that none exists. We can prove, at least, that if such example does exist, it would have a fairly complicated structure.'' Theorem A + B. Suppose that \(G\) is solvable and that some irreducible character of \(G\) factors as a product of two faithful characters \(\alpha\), \(\beta\). If \(G\) is not cyclic, then its Fitting height is at least 4. Next, \(\alpha(1)\) and \(\beta(1)\) have at least two common prime factors. Also, \(G\) has a noncentral minimal normal subgroup, and if \(M\) is any such subgroup, say \(p\)-subgroup, then \(p\) divides \(\alpha(1)\), \(\beta(1)\), and the rank of \(M\). Furthermore, the group of automorphisms of \(M\) induced by \(G\) is not \(p\)-nilpotent. The following module-theoretic conjecture is connected closely with the problem considered. Conjecture C. Let \(V\) be an irreducible \(FG\)-module, where \(F\) is a field. Suppose that we can factor \(G=XY\), where \(X\) and \(Y\) are subgroups of \(G\), and each of \(X\) and \(Y\) has a nonzero fixed point in \(V\). Then \(G\) acts trivially on \(V\). Theorem D. Assume that Conjecture C holds for some solvable group \(G\). If \(G\) has an irreducible character \(\alpha\beta\) with faithful characters \(\alpha\) and \(\beta\), then \(G\) is cyclic. Some intermediate results are proved. For example, Lemma (Lemma 2.1 + Corollary 2.2): Let \(\alpha,\beta\in\text{Irr}(G)\) and \(\alpha\beta\) is irreducible. (a) If \(\alpha\) is induced from some subgroup \(X\leq G\), then \(\beta_X\) is irreducible. Also, if \(\beta\) is induced from \(Y\leq G\), then \(G=XY\). (b) Every cyclic normal subgroup of \(G\) is central.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 1423891
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: faithful characters
zbMATH Keywords: minimal normal subgroups
zbMATH Keywords: irreducible modules
zbMATH Keywords: solvable groups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: jabr.1999.8121
OpenAlex ID: W2115397179
cites work: A characteristic class of characters of finite \(\pi\)-separable groups
cites work: Characters of Solvable and Symplectic Groups
cites work: Characters of \(\pi\)-separable groups
cites work: Q4312077
cites work: Irreducible products of characters in \(A_ n\)
DOI: 10.1006/JABR.1999.8121
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description: scientific article; zbMATH DE number 1423891