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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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