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A New Uniqueness Proof for the Held Group

A New Uniqueness Proof for the Held Group LEONARD H. SOICHER Introduction Let G be a finite simple group having an involution centralizer isomorphic to that of a 2-central involution in the Mathieu group M . Held [4] shows that G is 2i 10 3 2 3 isomorphic to M , L (2), or a simple group of order 2 .3 .5 .7 .17. We call the last 2i 5 possibility a Held group. In 1969, under the assumption that a Held group contains a subgroup 5 (4):2, G. Higman constructed a presentation satisfied by generators of Aut (He), for any Held group He. Computer coset enumeration was then used by J. McKay to show that the U 3 2 3 group H* thus presented has order 2 .3 .5 .7 .17, and so shows that there is at most one Held group containing an ,S (4): 2. Further work by Higman and McKay shows that the commutator subgroup of H* is in fact a Held group, thus settling the question of existence. Unfortunately, the work of Higman and McKay was never published, except for the presentation of H* (see [7]). Moreover, the existence of a subgroup iS (4): 2 of any Held group has only recently http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

A New Uniqueness Proof for the Held Group

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/23.3.235
Publisher site
See Article on Publisher Site

Abstract

LEONARD H. SOICHER Introduction Let G be a finite simple group having an involution centralizer isomorphic to that of a 2-central involution in the Mathieu group M . Held [4] shows that G is 2i 10 3 2 3 isomorphic to M , L (2), or a simple group of order 2 .3 .5 .7 .17. We call the last 2i 5 possibility a Held group. In 1969, under the assumption that a Held group contains a subgroup 5 (4):2, G. Higman constructed a presentation satisfied by generators of Aut (He), for any Held group He. Computer coset enumeration was then used by J. McKay to show that the U 3 2 3 group H* thus presented has order 2 .3 .5 .7 .17, and so shows that there is at most one Held group containing an ,S (4): 2. Further work by Higman and McKay shows that the commutator subgroup of H* is in fact a Held group, thus settling the question of existence. Unfortunately, the work of Higman and McKay was never published, except for the presentation of H* (see [7]). Moreover, the existence of a subgroup iS (4): 2 of any Held group has only recently

Journal

Bulletin of the London Mathematical SocietyWiley

Published: May 1, 1991

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