- journal article metadata
European Mathematical Society Publishing House
2018-06-09 23:30:01
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
139
2018
0
On $H_{\sigma}$-permutably embedded subgroups of finite groups
Wenbin
Guo
University of Science and Technology of China, Hefei, Anhui, China
Chi
Zhang
University of Science and Technology of China, Hefei, Anhui, China
Alexander
Skiba
Francisk Skorina Gomel State University, Gomel, Belarus
D.
Sinitsa
Francisk Skorina Gomel State University, Gomel, Belarus
Finite group, $\sigma$-Hall subgroup, $\sigma$-subnormal subgroup, $\sigma$-nilpotent group, $H_{\sigma}$-permutably embedded subgroup
Let $G$ 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 $\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$, $\sigma (G) =\sigma (|G|)$. 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 ${\mathcal cal H}$ contains exact one Hall $\sigma _{i}$-subgroup of $G$ for every $\sigma _{i}\in \sigma (G)$. A subgroup $A$ of $G$ is called (i) a $\sigma$-Hall subgroup of $G$ if $\sigma (A) \cap \sigma (|G:A|)=\emptyset$; (ii) ${\sigma}$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set ${\mathcal H}$ such that $AH^x=H^xA$ for all $H\in {\mathcal H}$ and all $x\in G$. We say that a subgroup $A$ of $G$ is $H_{\sigma}$-permutably embedded in $G$ if $A$ is a ${\sigma}$-Hall subgroup of some ${\sigma}$-permutable subgroup of $G$. We study finite groups $G$ having an $H_{\sigma}$-permutably embedded subgroup of order $|A|$ for each subgroup $A$ of $G$. Some known results are generalized.
Group theory and generalizations
General
143
158
10.4171/RSMUP/139-4
http://www.ems-ph.org/doi/10.4171/RSMUP/139-4
6
6
2018