Extremely closed subgroups and a variant on Glauberman’s Z∗-theorem

Let G be a finite group and let H be a subgroup of G.We say thatIn this paper, we determine the structure of finite groups with an extremely closed abelian p-subgroup for some prime p.In particular, we show that if G contains such a subgroup H, then G = N G (H)O p ′ (G).