ZJ-subgroup

Definition

The ZJ-subgroup is a subgroup-defining function for groups of prime power order, that sends a given group $P$ to the group defined in the following equivalent ways:

1. The center of the join of abelian subgroups of maximum order in $P$.
2. The intersection of the abelian subgroups of maximum order.

The term ZJ-functor is also used for this because the subgroup-defining function is a characteristic p-functor: it always returns a nontrivial characteristic subgroup for any nontrivial group of prime power order.

The letters ZJ are used because Z denotes the center and J denotes the join of abelian subgroups of maximum order (also called the Thompson subgroup). The result of applying the ZJ-functor to a group $P$ is denoted $ZJ(P)$ or $Z(J(P))$.

Relation with other subgroup-defining functions

Related functions

• ZJe-subgroup