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Minimal Presentations for Finite Groups

Minimal Presentations for Finite Groups J. W. WAMSLEY Suppose G is a group generated by a set of elements {a ..., a }, let F be the free lf n group on the set X = {x ..., x }, then there is a homomorphism, <f>, from F onto lf n G defined by (x ) (j) = a . Denote the kernel of this homomorphism by R, then t t G ^ F/R and we say G = F/R is a presentation for G. If we now select a set {/?!, ..., R }, of elements of R, such that .R is the normal closure of {R ..., R } in m ly m F , then R is defined by the set {R ..., i? } and we write the presentation for G as lt m G = {A- ..., x | i? ..., .R } = F/i?. In the case n and w are finite then we say G 1} n l 9 m is finitely presented. In particular any finite group, G, has a finite presentation as we may take the set X to be the whole of G = {g ...,&.} where \G\ = r, and if m 1 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Minimal Presentations for Finite Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/5.2.129
Publisher site
See Article on Publisher Site

Abstract

J. W. WAMSLEY Suppose G is a group generated by a set of elements {a ..., a }, let F be the free lf n group on the set X = {x ..., x }, then there is a homomorphism, <f>, from F onto lf n G defined by (x ) (j) = a . Denote the kernel of this homomorphism by R, then t t G ^ F/R and we say G = F/R is a presentation for G. If we now select a set {/?!, ..., R }, of elements of R, such that .R is the normal closure of {R ..., R } in m ly m F , then R is defined by the set {R ..., i? } and we write the presentation for G as lt m G = {A- ..., x | i? ..., .R } = F/i?. In the case n and w are finite then we say G 1} n l 9 m is finitely presented. In particular any finite group, G, has a finite presentation as we may take the set X to be the whole of G = {g ...,&.} where \G\ = r, and if m 1

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jul 1, 1973

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