# On Strongly Π-Permutable Subgroups of a Finite Group

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## Abstract

Let *σ* = {*σ*_{i} | *i* ∈ *I*} be some partition of the set of all primes ℙ,let ∅ ≠ Π ⊆ *σ*, and let *G* be a finite group. A set *ℋ* of subgroups of *G* is said to be a complete Hall Π-set of *G* if each member ≠ 1 of *ℋ* is a Hall *σ*_{i}-subgroup of *G* for some *σ*_{i} ∈ Π and *ℋ* has exactly one Hall *σ*_{i}-subgroup of *G* for every *σ*_{i} ∈ Π such that *σ*_{i} ∩ *π*(*G*) ≠ ∅. A subgroup *A* of *G* is called (i) Π-permutable in *G* if *G* has a complete Hall Π-set *ℋ* such that *AH*^{x} = *H*^{x}*A* for all *H* ∈ *ℋ* and *x* ∈ *G*; (ii) *σ*-subnormal in *G* if there is a subgroup chain *A* = *A*_{0} ≤ *A*_{1} ≤ ⋯ ≤ *A*_{t} = *G* such that either *A*_{i−1} ≤ *A*_{i} or *A*_{i}/(*A*_{i−1})*A*_{i} is a *σ*_{k}-group for some *k* for all *i* = 1,…,*t*; and (iii) strongly Π-permutable if *A* is Π-permutable and *σ*-subnormal in *G*. We study the strongly Π-permutable subgroups of *G*. In particular, we give characterizations of these subgroups and prove that the set of all strongly Π-permutable subgroups of *G* forms a sublattice of the lattice of all subgroups of *G*.

## Keywords

finite group subgroup lattice*σ*-subnormal subgroup strongly Π-permutable subgroup

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