On complete Hall $${\mathfrak {F}}$$-sets of subgroups of finite groups and $${\fancyscript{Z}}$$-permutability

Abstract

Let $${\mathfrak {F}}$$ be a class of finite groups and let G be a finite group. Assume that $$\sigma = \{\sigma _i : i \in I\}$$ is a partition of the set of prime numbers $${\mathbb {P}}$$ . A set $${\fancyscript{Z}}$$ of subgroups of G is called a complete Hall $${\mathfrak {F}}$$ -set of subgroups of G of type $$\sigma $$ if (i) for every $$W \in {\fancyscript{Z}}$$ , $$W \in {\mathfrak {F}}$$ and W is a Hall $$\sigma _i$$ -subgroup of G for some $$i \in I$$ and if (ii) for every $$\sigma _i \in \sigma $$ , $${\fancyscript{Z}}$$ contains exactly one and only one Hall $$\sigma _i$$ -subgroup of G. A subgroup H of G is said to be $${\fancyscript{Z}}$$ -permutable in G if H permutes with every member of $${\fancyscript{Z}}$$ . We investigate the influence of $${\fancyscript{Z}}$$ -permutable subgroups on the structure of finite groups. Also, we study some properties of $${\fancyscript{Z}}$$ -permutable subgroups. Several results from the literature are improved and generalized.