A reduction theorem for a conjecture on products of two π-decomposable groups

JOURNAL OF ALGEBRA

卷:379

Article

Kazarin, L. S.¹  Martinez-Pastor, A.²  Perez-Ramos, M. D.³ 

[1] Yaroslavl P Demidov State Univ, Dept Math, Yaroslavl 150000, Russia

[2] Univ Politecn Valencia, IUMPA UPV, Inst Univ Matemat Pura & Aplicada, Escuela Tecn Super Ingn Informat, Valencia 46022, Spain

[3] Univ Valencia, Dept Algebra, Burjassot, Valencia, Spain

关键词: Finite groups;    pi-Structure;    pi-Decomposable groups;    Products of subgroups;    Hall subgroups;   

DOI  :  10.1016/j.jalgebra.2013.01.017

来源: Elsevier

【 摘 要 】

For a set of primes pi, a group X is said to be pi-decomposable if X = X-pi x X-pi' is the direct product of a pi-subgroup X-pi and a pi'-subgroup X-pi', where pi' is the complementary of pi in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: Let pi be a set of odd primes. If the finite group G = AB is a product of two pi-decomposable subgroups A = A(pi) x A(pi') and B = B-pi x B-pi', then A(pi)B(pi) = B(pi)A(pi) and this is a Hall pi-subgroup of G. We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free