Characteristic p-functor

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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.