Central functor

This article defines a particular kind of map (functor) from a set of subgroups of a group to a set (possibly the same set) of subgroups

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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History

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

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 $H$, $V(H)\leq Z(H)$.
• For any $p$-subgroup $H$, and any $xinG$, $W(H^{x})=(W(H))^{x}$.
• If $Z(H)\leq Z(K)$ then $V(H)\leq V(K)$.

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.