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.