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.
VIEW: Definitions built on this | Facts about this: (facts closely related to Central functor, all facts related to Central functor) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions

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