Some classes of finite groups and mutually permutable products

This paper has been published in Journal of Algebra, 319(8):3343-3351 (2008). Copyright 2008 by Elsevier. http://dx.doi.org/10.1016/j.jalgebra.2007.12.001[EN] This paper is devoted to the study of mutually permutable products of finite groups. A factorised group G=AB is said to be a mutually permutable product of its factors A and B when each factor permutes with every subgroup of the other factor. We prove that mutually permutable products of Y-groups (groups satisfying a converse of Lagrange's theorem) and SC-groups (groups whose chief factors are simple) are SC-groups, by means of a local version. Next we show that the product of pairwise mutually permutable Y-groups is supersoluble. Finally, we give a local version of the result stating that when a mutually permutable product of two groups is a PST-group (that is, a group in which every subnormal subgroup permutes with all Sylow subgroups), then both factors are PST-groups.The second and the fourth authors have been supported by the Grant MTM2004-08219-C02-02 from MEC (Spain) and FEDER (European Union).Asaad, M.; Ballester Bolinches, A.; Beidleman, JC.; Esteban Romero, R. (2008). Some classes of finite groups and mutually permutable products. Journal of Algebra. 8(319). doi:10.1016/j.jalgebra.2007.12.001831

On infinite core-finite groups

A group G is CF (core-finite) if H/H_G is finite for all subgroups H of G. Here we introduce some special subclasses of the class of CF-groups and consider whether locally graded groups in these classes are abelian-by-finite

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Locally finite groups all of whose subgroups are boundedly finite over their cores

For n a positive integer, a group G is called core-n if H/H_G has order at most n for every subgroup H of G (where H_G is the normal core of H, the largest normal subgroup of G contained in H). It is proved that a locally finite core-n group G has an abelian subgroup whose index in G is bounded in terms of n