On One Open Question of the Theory of $\sigma $-Properties of a Finite Group

Abstract

Let $\sigma =\{\sigma _{i} \mid i\in I\}$ be some partition of the set of all primes and G a finite group. A subgroup A of G is $\sigma $-permutable in G provided G is $\sigma $-full; that is, G has a Hall $\sigma _{i}$-subgroup for all $i\in I$ and A permutes with all such Hall subgroups H of G; that is, $AH=HA$. Answering the Question 6.4 in Skiba (Probl Phys Math Tech 42(21):89–96, 2014), we get a description of finite $\sigma $-full groups G in which $\sigma $-permutability is a transitive relation.

Keywords

Finite group / $\sigma $-soluble group')">$\sigma $-soluble group / $\sigma $-complex')">Weak Robinson $\sigma $-complex / $\sigma $-permutable subgroup')">$\sigma $-permutable subgroup / $P\sigma T$-group')">$P\sigma T$-group

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Funding

National Natural Science Foundation of China(12171126)

Belarusian Republican Foundation for Fundamental Research(F23RNF-237)

Ministry of Education of the Republic of Belarus(20211328)