1 results
PRIMITIVE SUBGROUPS AND PST-GROUPS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 3 / June 2014
- Published online by Cambridge University Press:
- 18 July 2013, pp. 373-378
- Print publication:
- June 2014
-
- Article
-
- You have access
- Export citation
-
All groups considered in this paper are finite. A subgroup
$H$ of a group 
$G$ is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of 
$G$ containing 
$H$ as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of 
$G$ has index a power of a prime if and only if 
$G/ \Phi (G)$ is a solvable PST-group. Let 
$\mathfrak{X}$ denote the class of groups 
$G$ all of whose primitive subgroups have prime power index. It is established here that a group 
$G$ is a solvable PST-group if and only if every subgroup of 
$G$ is an 
$\mathfrak{X}$-group.
