Conjugacy functor: Difference between revisions

Revision as of 18:06, 24 March 2008

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 $x\in G$ , $W(H^{x})=(W(H))^{x}$ .

Examples

Examples of conjugacy functors include the identity mapping and the Thompson subgroup functor. Note also that any central functor is a conjugacy functor.

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}

@@ Line 11: / Line 11: @@
 ===Definition with symbols===
-Let <math>G</math> be a group and <math>p</math> a prime. A '''conjugacy functor''' is a map <math>W</math> from the collection of <math>p</math>-subgroups of <math>G</math> to the collection of <math>p</math>-subgroups of <math>G</math> that satisfies:
+Let <math>G</math> be a group and <math>p</math> a prime. A '''conjugacy functor''' is a map <math>W</math> from the collection of nontrivial <math>p</math>-subgroups of <math>G</math> to the collection of nontrivial <math>p</math>-subgroups of <math>G</math> that satisfies:
 * For any <math>p</math>-subgroup <math>H</math>, <math>W(H) \le H</math>.