Central functor

Origin of the term
The term was first used in the paper Transfer and fusion in finite groups by Finkel 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 central functor is a map $$V$$ from the collection of $$p$$-subgroups of $$G$$ to the collection of $$p$$-subgroups of $$G$$ that satisfies:


 * For any $$p$$-subgroup $$H$$, $$V(H) \le Z(H)$$.
 * For any $$p$$-subgroup $$H$$, and any $$x in G$$, $$W(H^x) = (W(H))^x$$.
 * If $$Z(H) \le Z(K)$$ then $$V(H) \le V(K)$$.

Clearly, every central functor is a conjugacy functor.

Examples
Examples of central functor include the center and the $$p^{th}$$ powers of elements in the center.