Conjugacy 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

History

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

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 $H$, $W(H)\leq H$.
• 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.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 288, Chapter 8 (p-constrained and p-stable groups), Section 4 (groups with subgroups of Glauberman type), ^{More info}

Journal references

• Transfer and fusion in finite groups by Jonathan Lazare Alperin and Daniel Gorenstein, Journal of Algebra, ISSN 00218693, Volume 6, Page 242 - 255(Year 1967): This paper discusses the normalizers of subgroups of a Sylow subgroup in a finite group, using the ideas of a conjugation family and Alperin's fusion theorem^{Weblink (hosted on ScienceDirect)More info}