The Influence of Weakly \({\sigma }\)-Permutably Embedded Subgroups on the Structure of Finite Groups

Abstract

Let \(\sigma =\{{\sigma _i|i\in I}\}\) be some partition of the set of all primes \({\mathbb {P}}\), G a finite group and \(\sigma (G)=\{{\sigma _i|\sigma _i \cap \pi (G) \ne \emptyset }\}\). A set \({\mathcal {H}} \) of subgroups of G is said to be a complete Hall \(\sigma \)-set of G if every non-identity member of \({\mathcal {H}}\) is a Hall \(\sigma _i\)-subgroup of G and \({\mathcal {H}}\) contains exactly one Hall \(\sigma _i\)-subgroup of G for every \(\sigma _i\in \sigma (G)\). G is said to be \(\sigma \)-full if G possesses a complete Hall \(\sigma \)-set. A subgroup H of G is said to be \(\sigma \)-permutable in G provided there is a complete Hall \(\sigma \)-set \({\mathcal {H}}\) of G such that \(HA^x=A^xH\) for all \(A\in {\mathcal {H}}\) and all \(x\in G\); \(\sigma \)-permutably embedded in G if H is \(\sigma \)-full and for every \(\sigma _i \in \sigma (H)\), every Hall \(\sigma _i\)-subgroup of H is also a Hall \(\sigma _i\)-subgroup of some \(\sigma \)-permutable subgroup of G. We call that a subgroup H of G is weakly \({\sigma }\)-permutably embedded in G if there exists a \(\sigma \)-subnormal subgroup T of G such that \(G=HT\) and \(H\cap T\le H_{\sigma eG}\), where \(H_{\sigma eG}\) is the subgroup of H generated by all those subgroups of H which are \(\sigma \)-permutably embedded in G. In this paper, we study the structure of G under the condition that some given subgroups of G are weakly \({\sigma }\)-permutably embedded in G. Some known results are generalized.

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Correspondence to Chi Zhang.

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Research was supported by the NNSF of China (11771409) and Anhui Initiative in Quantum Information Technologies (No. AHY150200)

Communicated by Hamid Mousavi.

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Hussain, M.T., Cao, C. & Zhang, C. The Influence of Weakly \({\sigma }\)-Permutably Embedded Subgroups on the Structure of Finite Groups. Bull. Iran. Math. Soc. 46, 1341–1356 (2020). https://doi.org/10.1007/s41980-019-00328-x

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Keywords

  • Finite groups
  • \(\sigma \)-Permutable subgroup
  • \(\sigma \)-Permutably embedded subgroup
  • Weakly \({\sigma }\)-permutably embedded subgroup
  • Supersoluble group

Mathematics Subject Classification

  • 20D10
  • 20D15
  • 20D20
  • 20D35