# On Strongly Π-Permutable Subgroups of a Finite Group

## 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

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Skiba A. N., “On some results in the theory of finite partially soluble groups,” Commun. Math. Stat., vol. 4, no. 2, 281–309 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Skiba A. N., “On
*σ*-subnormal and*σ*-permutable subgroups of finite groups,” J. Algebra, vol. 436, 1–16 (2015).MathSciNetCrossRefGoogle Scholar - 3.Skiba A. N., “Some characterizations of finite
*σ*-soluble*PσT*-groups,” J. Algebra, vol. 495, 114–129 (2018).MathSciNetCrossRefGoogle Scholar - 4.Guo W. and Skiba A. N., “On Π-quasinornial subgroups of finite groups,” Monatsh. Math., vol. 185, 443–453 (2018).MathSciNetCrossRefGoogle Scholar
- 5.Kegel O. H., “Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z., Bd 78, 205–221 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Ballester-Bolinches A., Esteban-Romero R., and Asaad M.,
*Products of Finite Groups*, Walter de Gruyter, Berlin and New York (2010).CrossRefzbMATHGoogle Scholar - 7.Ballester-Bolinches A. and Esteban-Romero R., “On finite soluble groups in which Sylow permutability is a transitive relation,” Acta Math. Hungar., vol. 101, 193–202 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Skiba A. N., “On finite groups for which the lattice of S-permutable subgroups is distributive,” Arch. Math., vol. 109, 9–17 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
- 9.Doerk K. and Hawkes T.,
*Finite Soluble Groups*, Walter de Gruyter, Berlin and New York (1992).CrossRefzbMATHGoogle Scholar - 10.Kimber T., “Modularity in the lattice of Σ-permutable subgroups,” Arch. Math., vol. 83, 193–203 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Schmidt R.,
*Subgroup Lattices of Groups*, Walter de Gruyter, Berlin (1994).CrossRefzbMATHGoogle Scholar