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:

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

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.

Relation with other subgroup-defining functions

Smaller subgroup-defining functions

• Center
• D*e-subgroup

Larger subgroup-defining functions

• ZJ-subgroup

Facts

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