Finite groups with abnormal or formational subnormal primary subgroups

Abstract

A subgroup H of a group G is abnormal if $x\in 〈H,H^x〉$ for every $x\in G$. A group is primary if its order is equal to a power of a prime. We indicate the structure of a finite group in which primary cyclic subgroups are abnormal or subnormal. We investigate finite groups with abnormal or formational subnormal primary subgroups for a subgroup-closed saturate lattice formation that contained all nilpotent subgroups. We also describe the structure of a group G in which every subgroup is abnormal or $\mathbb{P}$-subnormal. In particular, G has a generalized Sylow tower and every non-abnormal subgroup of G is supersoluble.

Keywords:

• Abnormal subgroup
• finite groups
• primary subgroup
• subnormal subgroup

Mathematics Subject Classification:

• 20D10
• 20D35