Enumerating representations in finite wreath products
type: Item
instance of: scholarly article
title: Enumerating representations in finite wreath products
zbMATH Open document ID: 1004.20016
published in: Advances in Mathematics
publication date: 2002-11-11T00:00:00Z
review text: Let \(\{\Pi_n\}\) be either a sequence of alternating groups \(\{A_n\}\) or of symmetric groups \(\{S_n\}\). Further, let \(H\) be a finite group, \(\Gamma\) a group and \(N\) a normal subgroup of \(H\) such that \(H/N\) is Abelian. Let \[ (H\wr\Pi_n)_N:=\{(f,\pi)\in H\wr\Pi_n:\prod_if(i)\in N\},\;n\geq 0. \] The main result of the paper is computing the exponential generating function of the sequence \[ \{|\Hom(\Gamma,(H\wr\Pi_n)_N)|\}_{n=0}^\infty \] and of certain refinements of it when \((H:N)\leq 2\) and \(\Gamma\) satisfies some weak finiteness conditions. Also some special cases of the main result related to symmetric and alternating wreath products and the Weyl group of type \(D_n\) are considered. Finally, using some earlier results of the author [Combinatorica 17, No. 4, 523-554 (1997; Zbl 0906.05002)], a complete asymptotic expansion of the function \(\{|\Hom(G,H\wr S_n)|\}\) for arbitrary finite groups \(G\) and \(H\) is obtained. Among others, these results generalize those of \textit{N. Chigira} [J. Algebra 180, No. 3, 653-661 (1996; Zbl 0854.05006)] concerning the solutions of the equation \(x^n=1\) in the groups \(H\wr A_n\), \(H\wr S_n\) and \((H\wr\Pi_n)_N=W_n\).
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E34
zbMATH DE Number: 1491733
zbMATH Keywords: wreath products
zbMATH Keywords: generating functions
zbMATH Keywords: alternating groups
zbMATH Keywords: symmetric groups
zbMATH Keywords: Weyl groups
author: T. W. Müller
reviewed by: Piroska Lakatos
MaRDI profile type: Publication
OpenAlex ID: W1996320647
cites work: The solutions of \(x^ d=1\) in finite groups
cites work: On Recursions Connected With Symmetric Groups I
cites work: Q5818508
cites work: Schreier systems in free products
cites work: Q3219268
cites work: Q3994461
cites work: Decomposable functors and the exponential principle
cites work: Subgroups of Finite Index in Free Groups
cites work: On Schreier Systems in Free Groups
cites work: Q4891333
cites work: Combinatorial Aspects of Finitely Generated Virtually Free Groups
cites work: Counting free subgroups of finite index
cites work: Finite group actions and asymptotic expansion of \(e^{P(z)}\)
cites work: Subgroup growth of free products
cites work: On Solutions of xd = 1 In Symmetric Groups
cites work: The asymptotics of 𝑒^{𝑃(𝑧)} and the number of elements of each order in 𝑆_{𝑛}
DOI: 10.1006/AIMA.1998.1885
description: scientific article; zbMATH DE number 1491733