University of Illinois Urbana-Champaign

On an analogue to the Grushko-Neumann theorem involving finite groups

Dabrowski, Walter Casimir

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Description

Title
On an analogue to the Grushko-Neumann theorem involving finite groups
Author(s)
Dabrowski, Walter Casimir
Issue Date
1996
Doctoral Committee Chair(s)
Rotman, Joseph J.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
Type of Resource
text
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http://hdl.handle.net/2142/21391
Copyright and License Information
Copyright 1996 Dabrowski, Walter Casimir

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