On equations in finite groups and invariants of representations for their subgroups
type: Item
instance of: scholarly article
title: On equations in finite groups and invariants of representations for their subgroups
zbMATH Open document ID: 0854.20011
DOI: 10.1016/0022-4049(95)00071-2
author: S. P. Strunkov
published in: Journal of Pure and Applied Algebra
publication date: 1997-01-20T00:00:00Z
review text: This paper is devoted to certain questions in group representation theory from the viewpoint of the theory of equations of groups. The main part is dedicated to the study of the existence of \(p\)-blocks of defect 0. Let \(G\) be a finite group, let a prime \(p\) divide \(|G|\) and let \(\chi\) be an irreducible complex character of \(G\). If \(|G|=p^cm\), \(\chi(1)=p^{c_1}m_1\) with \((m,p)=(m_1,p)=1\), then the nonnegative integer \(c-c_1\) is called the \(p\)-defect of \(\chi\). To give the flavour of the subject, we quote the following result and its corollary. Theorem. Let \(f(x_1,\dots,x_k,u_1,\dots,u_e)\) be a function on a finite group \(G\) and let \(f\) be a product of elementary functions \([x_i,x_{i+1}]\) and \(u^{x_s}_j\) (\(x_i\in G\), \(u_j\in G_p\), \(G_p\) is a Sylow \(p\)-subgroup of \(G\)). Moreover, variables that enter in different elementary factors, are distinct and \(k \geq 2\). Then \(G\) has a \(p\)-block of defect 0 if and only if for some \(g\in G\) the number of solutions of the equation \(f(x_1,\dots,x_k,u_1,\dots,u_e)=g\) does not divide \(p |G_p|^e\). Corollary. The following conditions are equivalent: (i) A finite group \(G\) has a \(p\)-block of defect 0. (ii) The number of solutions of the equation \([x,y]=g\) is prime to \(p\) for some \(g\in G\) \((x,y\in G)\). (iii) The number of solutions of the equation \(u^x v^y=g\) does not divide \(p|G_p|^2\) for some \(g\in G\) (\(x,y\in G\), \(u,v\in G_p\)). (iv) Let \(k\) be a natural number. Then the number of solutions of the equation \([x_1,x_2][x_3,x_4]\cdots[x_{2k-1},x_{2k}]=g\) (\(x_1,\dots,x_{2k}\in G\)) is prime to \(p\) for some \(g\in G\).
reviewed by: G.Karpilovsky
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 908495
zbMATH Keywords: equations of groups
zbMATH Keywords: existence of \(p\)-blocks
zbMATH Keywords: finite groups
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: \(p\)-defect
zbMATH Keywords: product of elementary functions
zbMATH Keywords: Sylow \(p\)-subgroups
zbMATH Keywords: number of solutions
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00071-2
OpenAlex ID: W2091176831
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description: scientific article; zbMATH DE number 908495