Central functor

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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) \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.