Two subgroups X and Y of a group G are said to be conditionallyi.e., X Y g is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.
Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Y g for some g ∈ X, Y , for all X A and Y B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.
Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) if X permutes with Y g for some g ∈ X, Y , for all X ≤ A and all Y ≤ B. In this paper we study finite products of tcc-permutable subgroups, focussing mainly on structural properties of such products. As an application, new achievements in the context of formation theory are obtained.
A lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving ^-subnormal subgroups, for a lattice formation & of full characteristic, are studied. For a subgroup-closed saturated formation #', a characterisation of the Sf-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the
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