Finite groups with permutable Hall subgroups

Abstract

Let \sigma =\{{{\displaystyle \sigma }}_i|i\in I\} be a partition of the set of all primes \mathbb{P}, and let G be a finite group. A set \mathcal{H} of subgroups of G is said to be a complete Hall\sigma-set of G if every member \ne \mathrm{1} of H is a Hall σi-subgroup of G for somei ∈ I and H contains exactly one Hall σi-subgroup of G for every i such that {{\displaystyle \sigma }}_i\cap \pi (G)\ne \phi. In this paper, we study the structure of G under the assuming that some subgroups of G permutes with all members of \mathcal{H} .

Key words： Finite group; Hall subgroup; complete Hall σ-set; permutable subgroup; supersoluble group

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