# ZJ-subgroup

## Definition

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

- The center of the join of abelian subgroups of maximum order in .
- 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 is denoted or .