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 PFMT, 2017, Issue 4(33), Pages 84–88 (Mi pfmt540)

MATHEMATICS

Finite groups with $H_\sigma$-subnormally embedded subgroups

D. A. Sinitsaa, A. N. Skibaa, W. Guob, Chi Zhangb

a F. Scorina Gomel State University
b University of Science and Technology of China, Hefei

Abstract: Пусть $G$ be a finite group. Let $\sigma=\{\sigma_i| i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $n$ an integer. We write $\sigma(n)=\{\sigma_i |\sigma_i\cap \pi(n)\ne\varnothing\}$, $\sigma(G)=\sigma(|G|)$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member of $\mathcal{H}\setminus\{1\}$ is a Hall $\sigma_i$-subgroup of $G$ for some $\sigma_i$ and $\mathcal{H}$ contains exact one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in\sigma(G)$. A subgroup $A$ of $G$ is called a $\sigma$-Hall subgroup of $G$ if $\sigma(|A|)\cap\sigma(|G:A|)=\varnothing$. We say that a subgroup $A$ of $G$ is $H_\sigma$-subnormally embedded in $G$ if $A$ is a $\sigma$-Hall subgroup of some $\sigma$-subnormal subgroup of $G$.

Keywords: finite group, $\sigma$-subnormal subgroup, $\sigma$-permutable subgroup, $\sigma$-Hall subgroup, $H_\sigma$-subnormally embedded subgroup.

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UDC: 512.542

Citation: D. A. Sinitsa, A. N. Skiba, W. Guo, Chi Zhang, “Finite groups with $H_\sigma$-subnormally embedded subgroups”, PFMT, 2017, no. 4(33), 84–88

Citation in format AMSBIB
\Bibitem{SinSkiGuo17} \by D.~A.~Sinitsa, A.~N.~Skiba, W.~Guo, Chi~Zhang \paper Finite groups with $H_\sigma$-subnormally embedded subgroups \jour PFMT \yr 2017 \issue 4(33) \pages 84--88 \mathnet{http://mi.mathnet.ru/pfmt540}