Statistics of isomorphism types in free products.
type: Item
instance of: scholarly article
title: Statistics of isomorphism types in free products.
zbMATH Open document ID: 1243.20033
author: T. W. Müller
author: Jan-Christoph Schlage-Puchta
published in: Advances in Mathematics
publication date: 2010-05-10T00:00:00Z
review text: Let \(\Gamma=G_1*G_2*\cdots*G_s*F_r\) be a free product of finite groups \(G_i\) and a free group \(F_r\) of rank \(r\) and let \(U_1,\dots,U_t\) be a complete list of isomorphism types of non-trivial subgroups of \(G_1,\dots,G_s\). Every finite-index subgroup \(\Delta\) of \(\Gamma\) is of the form \(\Delta\cong U_1^{*\lambda_1}*U_2^{*\lambda_2}*\cdots*U_t^{*\lambda_t}*F_\mu\) with non-negative integers \(\lambda_1,\dots,\lambda_t,\mu\). The tuple \((\lambda_1,\dots,\lambda_t,\mu)\) is called the isomorphism type of \(\Delta\). The first aim of the article is an algorithmic determination of the isomorphism type of a subgroup of finite index given by its coset representation. The authors determine the set of isomorphism types realized by finite-index subgroups, the asymptotics of the subgroup numbers with prescribed isomorphism types, and the distribution of the isomorphism types among subgroups of fixed index. Apart from group-theoretic arguments, the proofs make use of asymptotic, combinatorial, and probabilistic ideas and techniques.
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F69
zbMATH DE Number: 5706204
zbMATH Keywords: free products
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: isomorphism types of subgroups
zbMATH Keywords: representation types
zbMATH Keywords: numbers of subgroups
zbMATH Keywords: Klein realization problem
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: j.aim.2009.12.011
OpenAlex ID: W2068738493
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DOI: 10.1016/J.AIM.2009.12.011
description: scientific article; zbMATH DE number 5706204