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Finite Groups with Sylow 2‐Subgroups of Nilpotency Class 2

Finite Groups with Sylow 2‐Subgroups of Nilpotency Class 2 FINITE GROUPS WITH SYLOW 2-SUBGROUPS OF NILPOTENCY CLASS 2 A. R. CAMINA In [1] the author and T. M. Gagen proved that if G is a finite group whose Sylow 2-subgroup S has a normal subgroup T such that T and S/T are cyclic with \S/T\ ^ 4 then G is soluble. P. L. Chabot in his thesis [2] considered the problem of which 2-groups with cyclic derived groups could be the Sylow 2-subgroups of a finite simple group. The following result is implicitly contained in his thesis: if G is a finite group whose Sylow 2-subgroup S is two-generator with cyclic derived group S' and S/S' has n o direct factor of order 2 then G is soluble. The main purpose of this note is to prove a variation on this theme. THEOREM. Let G be a finite group with a Sylow 2-subgroup S which has nilpotency class at most 2 and S/S' has no direct factor of order 2. Then G is soluble. The proof depends heavily on the results of D. M. Goldschmidt [3] and I would like to quote here the relevant results from his thesis. THEOREM B (Goldschmidt). Suppose S is a Sylow 2-subgroup http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Finite Groups with Sylow 2‐Subgroups of Nilpotency Class 2

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/2.3.290
Publisher site
See Article on Publisher Site

Abstract

FINITE GROUPS WITH SYLOW 2-SUBGROUPS OF NILPOTENCY CLASS 2 A. R. CAMINA In [1] the author and T. M. Gagen proved that if G is a finite group whose Sylow 2-subgroup S has a normal subgroup T such that T and S/T are cyclic with \S/T\ ^ 4 then G is soluble. P. L. Chabot in his thesis [2] considered the problem of which 2-groups with cyclic derived groups could be the Sylow 2-subgroups of a finite simple group. The following result is implicitly contained in his thesis: if G is a finite group whose Sylow 2-subgroup S is two-generator with cyclic derived group S' and S/S' has n o direct factor of order 2 then G is soluble. The main purpose of this note is to prove a variation on this theme. THEOREM. Let G be a finite group with a Sylow 2-subgroup S which has nilpotency class at most 2 and S/S' has no direct factor of order 2. Then G is soluble. The proof depends heavily on the results of D. M. Goldschmidt [3] and I would like to quote here the relevant results from his thesis. THEOREM B (Goldschmidt). Suppose S is a Sylow 2-subgroup

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Nov 1, 1970

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