Characteristic p-functor

Definition
Let $$p$$ be a prime number. A characteristic $$p$$-functor $$W$$ associates to every finite $$p$$-group, a characteristic subgroup such that:


 * Given an isomorphism of $$p$$-groups $$\varphi:G \to H$$, $$\varphi$$ maps $$W(G)$$ to $$W(H)$$
 * If $$G$$ is nontrivial, $$W(G)$$ is nontrivial.

Characteristic $$p$$-functors are thus subgroup-defining functions restricted to $$p$$-groups, with a nontriviality condition. Note that sometimes, the nontriviality condition is emphasized by the use of the term positive, so that we say positive characteristic p-functor.

A characteristic $$p$$-functor gives rise to a conjugacy functor, and more generally, to a section conjugacy functor, for every $$p$$-group.