# 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.

## Relation with other subgroup-defining functions

### Smaller subgroup-defining functions

Subgroup-defining function Meaning Proof of containment Proof of strictness Conditions for equality
Center elements that commute with every element of the group D*-subgroup contains center center need not contain D*-subgroup In a group of nilpotency class two the center coincides with the D*-subgroup
D*e-subgroup use elementary abelian subgroups instead of abelian subgroups in the definition

### Larger subgroup-defining functions

Subgroup-defining function Meaning Proof of containment Proof of strictness
ZJ-subgroup center of the join of abelian subgroups of maximum order ZJ-subgroup contains D*-subgroup D*-subgroup need not contain ZJ-subgroup