On regular subgroup functors of finite groups

Abstract

Abstract A subgroup functor τ \tau is said Φ \Phi -regular if for all primitive groups G G , whenever H ∈ τ ( G ) H\in \tau \left(G) is a p p -subgroup and N N is a minimal normal subgroup of G G , then ∣ G : N G ( H ∩ N ) ∣ = p d | G:{N}_{G}\left(H\cap N)| ={p}^{d} for some integer d d . In this article, we investigate groups in which some primary subgroups are τ \tau -subgroups for a Φ \Phi -regular subgroup functor τ \tau , and we obtain new criteria for the supersolubility or p p -nilpotency of a group.