Pronormality and Strong Pronormality of Subgroups

Abstract

A subgroup H of a group G is said to be pronormal if, for any element g ∈ G, subgroups H and H g are conjugate in $$ \left\langle {H,{H^g}} \right\rangle $$ . A subgroup H of a group G is said to be strongly pronormal if, for any subgroup K ≤ H and any element g ∈ G, there exists an element $$ x\in \left\langle {H,{K^g}} \right\rangle $$ such that K gx ≤ H. Many known examples of pronormal subgroups, namely, normal subgroups, maximal subgroups, Sylow subgroups of finite groups, and Hall subgroups of finite soluble groups, will also exemplify strongly pronormal subgroups. It is shown that Carter subgroups of finite groups (which are always pronormal) are not strongly pronormal in general, even in soluble groups.