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.