On the residual of a factorized group with widely supersoluble factors

2020/02/15 by Monakhov, Victor S., Trofimuk, Alexander A.
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2002.06355

Abstract

Let \Bbb P be the set of all primes. A subgroup H of a group G is called \it \mathbb P-subnormal in G, if either H=G, or there exists a chain of subgroups H=H₀≤ H₁≤ … ≤ H_n=G, |H_i:H_i-1|∈ \Bbb P, ∀ i. A group G is called \it widely supersoluble, w-supersoluble for short, if every Sylow subgroup of G is \mathbb P-subnormal in G. A group G=AB with \mathbb P-subnormal w-supersoluble subgroups A and B is studied. The structure of its w-supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the A-residual of G. Here A is the formation of all groups with abelian Sylow subgroups. Besides, we obtain new sufficient conditions for the w-supersolubility of such group G.

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