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 (Formula presented) for all g ∈ G. 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 (Formula presented). This is a variant on the celebrated Glauberman’s Z^∗-theorem.