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ON $\unicode[STIX]{x1D70E}$-QUASINORMAL SUBGROUPS OF FINITE GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  12 November 2018

BIN HU ,
JIANHONG HUANG Open the ORCID record for JIANHONG HUANG [Opens in a new window]  and
ALEXANDER N. SKIBA
BIN HU
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, PR China email hubin118@126.com
JIANHONG HUANG*
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, PR China email jhh320@126.com
ALEXANDER N. SKIBA
Affiliation:
Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University, Gomel 246019, Belarus email alexander.skiba49@gmail.com
*
✉jhh320@126.com
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Abstract

Let $G$ be a finite group and $\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup _{i\in I}\unicode[STIX]{x1D70E}_{i}$ and $\unicode[STIX]{x1D70E}_{i}\cap \unicode[STIX]{x1D70E}_{j}=\emptyset$ for all $i\neq j$. We say that $G$ is $\unicode[STIX]{x1D70E}$-primary if $G$ is a $\unicode[STIX]{x1D70E}_{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: $\unicode[STIX]{x1D70E}$-subnormal in$G$ if there is a subgroup chain $A=A_{0}\leq A_{1}\leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\unlhd A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\unicode[STIX]{x1D70E}$-primary for all $i=1,\ldots ,n$; modular in$G$ if the following conditions hold: (i) $\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ for all $X\leq G,Z\leq G$ such that $X\leq Z$ and (ii) $\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ for all $Y\leq G,Z\leq G$ such that $A\leq Z$; and $\unicode[STIX]{x1D70E}$-quasinormal in$G$ if $A$ is modular and $\unicode[STIX]{x1D70E}$-subnormal in $G$. We study $\unicode[STIX]{x1D70E}$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $\unicode[STIX]{x1D70E}$-quasinormal in $G$, then every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ is $\unicode[STIX]{x1D70E}$-central in$G$, that is, the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is $\unicode[STIX]{x1D70E}$-primary.

Keywords

finite group𝜎-nilpotent group𝜎-subnormal subgroupmodular subgroup𝜎-quasinormal subgroup

MSC classification

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary: 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 413 - 420
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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Footnotes

Research is supported by an NNSF grant of China (Grant No. 11401264) and a TAPP of Jiangsu Higher Education Institutions (PPZY 2015A013).

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