Finite Groups with ${{{\mathcal {H}}}}$-Permutable Subgroups

Abstract

Let $\sigma =\{\sigma _{i}\ |\ i\in I\}$ be some partition of the set $\mathbb {P}$ of all primes and G a finite group. A set ${{{\mathcal {H}}}}$ of subgroups of G is said to be a complete Hall $\sigma $ -set of G if every member $\ne 1$ of ${{{\mathcal {H}}}}$ is a Hall $\sigma _{i}$-subgroup of G for some $i\in I$ and ${{\mathcal {H}}}$ contains exactly one Hall $\sigma _{i}$-subgroup of G for every i such that $\sigma _{i}\cap \pi (G)\ne \emptyset $. A subgroup A of G is said to be ${{\mathcal {H}}}$-permutable if A permutes with all members of the complete Hall $\sigma $-set ${{{\mathcal {H}}}}$ of G. In this paper, we study the structure of G under the assuming that some subgroups of G are ${{\mathcal {H}}}$-permutable.

Keywords

Finite group / Hall subgroup / $\sigma $-set')">Complete Hall $\sigma $-set / ${{{\mathcal {H}}}}$-permutable subgroup')">${{{\mathcal {H}}}}$-permutable subgroup / PST-group

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Funding

National Natural Science Foundation of China(11371335)

Chinese Academy of Sciences Visiting Professorship for Senior International Scientists(2010T2J12)