D*-subgroup

History
This group was introduced in a paper by George Glauberman and Ronald Solomon, pending publication as of 2012.

Definition
Let $$p$$ be a prime number and $$P$$ be a finite p-group. The $$D^*$$-subgroup of $$P$$, denoted $$D^*(P)$$, is defined as the unique maximal element in the collection $$\mathcal{D}^*(P)$$ of subgroups of $$P$$ defined as:

$$\mathcal{D}^*(P) = \{ A \le P \mid A \mbox{ is abelian and }\operatorname{class}(\langle A,x \rangle ) \le 2 \implies x \in C_P(A) \ \forall \ x \in P \}$$

Well definedness
The proof that this collection of subgroups has a unique maximal element follows from the observation that the property is a normalizing join-closed subgroup property and the fact that normalizing join-closed subgroup property in nilpotent group implies unique maximal element, along with the observation that prime power order implies nilpotent and that we are dealing with finite groups.

Facts

 * p-constrained and p-stable implies normalizer of D*-subgroup generates whole group with p'-core for odd p