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A GENERALISATION OF FINITE PT-GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 3 / June 2018
- Published online by Cambridge University Press:
- 07 March 2018, pp. 396-405
- Print publication:
- June 2018
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- Article
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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.
