# Characteristic p-functor

From Groupprops

## Definition

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

- Given an isomorphism of -groups , maps to
- If is nontrivial, is nontrivial.

Characteristic -functors are thus subgroup-defining functions restricted to -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 -functor gives rise to a conjugacy functor, and more generally, to a section conjugacy functor, for every -group.