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Title:On an analogue to the Grushko-Neumann theorem involving finite groups
Author(s):Dabrowski, Walter Casimir
Doctoral Committee Chair(s):Rotman, Joseph J.
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
Abstract:Let C be a class of finite groups closed under the operations of taking subgroups, quotients, and extensions. Let H and K be pro-C-groups and let $G=H*K$ be their free pro-C-product. An open question in the theory of profinite groups is whether or not $d(G)=d(H)+d(K),$ which is an analogue to the Grushko-Neumann theorem for abstract free products of groups.
Ribes and Wong show that the two following assertions are equivalent: (1) For every pair H and K of pro-C-groups, $d(H\ *\ K)=d(H)+d(K).$ (2) For every pair of H and K of finite groups in C, there is a finite group G in C such that $G={},\ {\rm and}\ d(G)=d(H)+d(K).$
Using the classification of finite simple groups, we provide evidence that assertion (2) in the case that C consists of all finite groups is false for certain H and K. This might be thought of as an extension of the evidence provided so far by Kovacs, Sim, and Lucchini.
Issue Date:1996
Type:Text
Language:English
URI:http://hdl.handle.net/2142/21391
ISBN:9780591086799
Rights Information:Copyright 1996 Dabrowski, Walter Casimir
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9702494
OCLC Identifier:(UMI)AAI9702494


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