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The Identification of Finite Groups of PSL (5, q)‐Type and PSU (5, q)‐Type

The Identification of Finite Groups of PSL (5, q)‐Type and PSU (5, q)‐Type THE IDENTIFICATION OF FINITE GROUPS OF PSL (5, q)-TY?E AND PSU (5, q)-TY?E MICHAEL J. COLLINSf AND RONALD M. SOLOMON 1. Introduction The Sylow 2-subgroups of the finite simple groups PSL (5, q) and PSU (5, q) for odd q fall into two classes depending on the congruence of q modulo 4. In an attempt to characterise these groups by the structure of their Sylow 2-subgroups [3], Mason has shown that a corefree group having one of these groups as Sylow 2-subgroup and having no subgroup of index 2 has a fusion pattern for its involutions and a 2-local structure which is similar to that in one of the actual groups. In particular, he was able to divide such groups into two categories according to the nonabelian composi- tion factors of the centralisers of involutions, leaving an indeterminacy over the cores and possible odd order quotients. With a definition which is given in Section 2 and analogously in Section 5, he showed that such a group was either of PSL (5, q)-type or of PSU (5, q)-type. In this paper, we shall identify these groups. THEOREM. Let q be an odd prime power, and let G be a finite http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Identification of Finite Groups of PSL (5, q)‐Type and PSU (5, q)‐Type

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/7.2.113
Publisher site
See Article on Publisher Site

Abstract

THE IDENTIFICATION OF FINITE GROUPS OF PSL (5, q)-TY?E AND PSU (5, q)-TY?E MICHAEL J. COLLINSf AND RONALD M. SOLOMON 1. Introduction The Sylow 2-subgroups of the finite simple groups PSL (5, q) and PSU (5, q) for odd q fall into two classes depending on the congruence of q modulo 4. In an attempt to characterise these groups by the structure of their Sylow 2-subgroups [3], Mason has shown that a corefree group having one of these groups as Sylow 2-subgroup and having no subgroup of index 2 has a fusion pattern for its involutions and a 2-local structure which is similar to that in one of the actual groups. In particular, he was able to divide such groups into two categories according to the nonabelian composi- tion factors of the centralisers of involutions, leaving an indeterminacy over the cores and possible odd order quotients. With a definition which is given in Section 2 and analogously in Section 5, he showed that such a group was either of PSL (5, q)-type or of PSU (5, q)-type. In this paper, we shall identify these groups. THEOREM. Let q be an odd prime power, and let G be a finite

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jul 1, 1975

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