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 that H is extremely closed in G if H, Hᵍ NG (H) =H for all g G. In 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=NG (H) O₏' (G). This is a variant on the celebrated Glauberman's Z^*-Theorem.