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Licensed Unlicensed Requires Authentication Published by De Gruyter July 19, 2019

On weakly 𝓗-permutable subgroups of finite groups

Chenchen Cao, Venus Amjid and Chi Zhang
From the journal Mathematica Slovaca

Abstract

Let σ = {σiiI} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σiσiπ(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H0H1 ≤ … ≤ Ht = G such that either Hi−1 is normal in Hi or Hi/(Hi−1)Hi is σ-primary for all i = 1, …, t. A set 𝓗 of subgroups of G is said to be a complete Hallσ-set of G if every non-identity member of 𝓗 is a Hall σi-subgroup of G for some i and 𝓗 contains exactly one Hall σi-subgroup of G for every σiσ(G). Let 𝓗 be a complete Hall σ-set of G. A subgroup H of G is said to be 𝓗-permutable if HA = AH for all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and HTH𝓗, where H𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable.

By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.


This work was supported by the NNSF of China (11771409), Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences and Anhui Initiative in Quantum Information Technologies (AHY150200)


  1. (Communicated by Vincenzo Marra )

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Received: 2017-09-05
Accepted: 2019-01-28
Published Online: 2019-07-19
Published in Print: 2019-08-27

© 2019 Mathematical Institute Slovak Academy of Sciences