Finite Groups in Which the Automizers of Some Abelian Subgroups are Trivial

Abstract

If H is a subgroup of a finite group G, then the automizer AutG(H) of H in G is defined as the group of automorphisms of H induced by conjugation by elements of NG(H). A finite group G is called an NNC-group if for any non-normal abelian subgroup A, either [Formula: see text] or [Formula: see text]. In this paper, classifications of nilpotent NNC-groups and non-solvable NNC-groups are given. We also investigate the solvable NNC-groups and describe the structure of solvable NNC-groups.