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 defining ingredient::center of the defining ingredient::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 $$Z(J(P))$$.

Related functions

 * ZJe-subgroup