ZJ-subgroup

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Template:Prime-parametrized subgroup-defining function

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