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

On weakly ๐“—-permutable subgroups of finite groups

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

Abstract

Let ฯƒ = {ฯƒi โˆฃi โˆˆ I} be some partition of the set of all primes โ„™, G be a finite group and ฯƒ(G) = {ฯƒiโˆฃฯƒi โˆฉ ฯ€(G) โ‰  โˆ…}. G is said to be ฯƒ-primary if โˆฃฯƒ(G)โˆฃ โ‰ค 1. A subgroup H of G is said to be ฯƒ-subnormal in G if there exists a subgroup chain H = H0 โ‰ค H1 โ‰ค โ€ฆ โ‰ค Ht = G such that either Hiโˆ’1 is normal in Hi or Hi/(Hiโˆ’1)Hi is ฯƒ-primary for all i = 1, โ€ฆ, t. A set ๐“— of subgroups of G is said to be a complete Hallฯƒ-set of G if every non-identity member of ๐“— is a Hall ฯƒi-subgroup of G for some i and ๐“— contains exactly one Hall ฯƒi-subgroup of G for every ฯƒi โˆˆ ฯƒ(G). Let ๐“— be a complete Hall ฯƒ-set of G. A subgroup H of G is said to be ๐“—-permutable if HA = AH for all A โˆˆ ๐“—. We say that a subgroup H of G is weakly ๐“—-permutable in G if there exists a ฯƒ-subnormal subgroup T of G such that G = HT and H โˆฉ T โ‰ค H๐“—, where H๐“— is the subgroup of H generated by all those subgroups of H which are ๐“—-permutable.

By using the weakly ๐“—-permutable subgroups, we establish some new criteria for a group G to be ฯƒ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.

  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

ยฉ 2019 Mathematical Institute Slovak Academy of Sciences