D*-subgroup

History

This group was introduced in a paper by George Glauberman and Ronald Solomon.

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:

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

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