3 research outputs found
A reduction theorem for a conjecture on products of two Âż-decomposable groups
[EN] 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.The second and third author have been supported by Proyecto MTM2010-19938-C03-02, Ministerio de Economia y Competitividad, Spain. The first author would like to thank the Universitat de Valencia and the Universitat Politecnica de Valencia for their warm hospitality during the preparation of this paper. He has been also supported by RFBR project 13-01-00469.Kazarin, LS.; MartĂnez Pastor, A.; Perez Ramos, MD. (2013). A reduction theorem for a conjecture on products of two Âż-decomposable groups. Journal of Algebra. 379:301-313. https://doi.org/10.1016/j.jalgebra.2013.01.017S30131337
On the product of a π-group and a π-decomposable group
[EN] The main result in the paper states the following: Let Ď€ be a set of odd primes. Let the finite group G=AB be the product of a Ď€ -decomposable subgroup A=OĎ€(A)Ă—Oπ′(A) and a Ď€ -subgroup B . Then OĎ€(A)â©˝OĎ€(G); equivalently the group G possesses Hall Ď€ -subgroups. In this case OĎ€(A)B is a Hall Ď€-subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G.The second and third authors have been supported by Proyecto MTM2004-06065-C02-02, Ministerio de EducaciĂłn y Ciencia and FEDER, Spain. The first author would like to thank the Universitat de València and the Universidad PolitĂ©cnica de Valencia for their warm hospitality during the preparation of this paper.Kazarin, L.; MartĂnez Pastor, A.; PĂ©rez-Ramos, M. (2007). On the product of a Ď€-group and a Ď€-decomposable group. Journal of Algebra. 315(2):640-653. doi:10.1016/j.jalgebra.2007.06.004S640653315