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A GENERALISATION OF FINITE PT-GROUPS
Published online by Cambridge University Press: 07 March 2018
Abstract
Let $G$ be a group and
$\unicode[STIX]{x1D70E}=\{\unicode[STIX]{x1D70E}_{i}\mid i\in I\}$ some partition of the set of all primes. A subgroup
$A$ of
$G$ is
$\unicode[STIX]{x1D70E}$-subnormal in
$G$ if there is a subgroup chain
$A=A_{0}\leq A_{1}\leq \cdots \leq A_{m}=G$ such that either
$A_{i-1}\unlhd A_{i}$ or
$A_{i}/(A_{i-1})_{A_{i}}$ is a finite
$\unicode[STIX]{x1D70E}_{j}$-group for some
$j=j(i)$ for
$i=1,\ldots ,m$, and it is modular in
$G$ if
$\langle X,A\cap Z\rangle =\langle X,A\rangle \cap Z$ when
$X\leq Z\leq G$ and
$\langle A,Y\cap Z\rangle =\langle A,Y\rangle \cap Z$ when
$Y\leq G$ and
$A\leq Z\leq G$. The group
$G$ is called
$\unicode[STIX]{x1D70E}$-soluble if every chief factor
$H/K$ of
$G$ is a finite
$\unicode[STIX]{x1D70E}_{i}$-group for some
$i=i(H/K)$. In this paper, we describe finite
$\unicode[STIX]{x1D70E}$-soluble groups in which every
$\unicode[STIX]{x1D70E}$-subnormal subgroup is modular.
- Type
- Research Article
- Information
- Copyright
- © 2018 Australian Mathematical Publishing Association Inc.
Footnotes
This research is supported by the NNSF of China (grant no. 11401264) and TAPP of Jiangsu Higher Education Institutions (PPZY 2015A013).
References
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