Conjugacy functor

Origin of the term
The term was first used in the paper Transfer and fusion in finite groups by Alperin and Gorenstein in the Journal of Algebra, 6 (1967), Pages 242-255.

Definition with symbols
Let $$G$$ be a group and $$p$$ a prime. A conjugacy functor is a map $$W$$ from the collection of nontrivial $$p$$-subgroups of $$G$$ to the collection of nontrivial $$p$$-subgroups of $$G$$ that satisfies:


 * For any $$p$$-subgroup $$H$$, $$W(H) \le H$$.
 * For any $$p$$-subgroup $$H$$, and any $$x \in G$$, $$xW(H)x^{-1} = W(xHx^{-1})$$.

Examples
Examples of conjugacy functors include the identity mapping, the functors corresponding to different possible Thompson subgroups, and the ZJ-functor.

Note also that any central functor is a conjugacy functor. Also, every characteristic p-functor is a conjugacy functor.