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

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 this 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

Facts

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