Convention:Conjugation: Difference between revisions

Revision as of 16:11, 27 March 2008

This article is about a convention that is followed in this wiki. The aim is that every page on the wiki follows this convention unless explicitly stated otherwise on the page; however, in practice, this may not have been implemented
Also see switching between the left and right action conventions for background on the differences between various conventions.

If $G$ is a group and $g\in G$ , the conjugation by $g$ is defined as the map:

$c_{g}:x\mapsto gxg^{-1}$

This convention is compatible with the convention that a group action is on the left.

This notation is followed in a number of pages.

However, the notation $x^{g}$ is used to denote conjugation on the right. In other words $x^{g}=g^{-1}xg$ . Thus, we have $(x^{g})^{h}=x^{gh}$ .

The latter notation is typically used in the theory of finite groups, when doing calculations involving conjugates and commutators. The primary advantage of this is that x^{gh} = (x^g)^h</math>.

@@ Line 7: / Line 7: @@
 This convention is compatible with [[Convention:Group action on left|the convention that a group action is on the left]].
-The conjugation of <math>x</math> by <math>g</math> is also sometimes denoted as <math>x^g</math>. Note that with the ''left'' convention, <math>(x^g)^h = x^{hg}</math> and not <math>x^{gh}</math>.
+This notation is followed in a number of pages.
+However, the notation <math>x^g</math> is used to denote conjugation on the right. In other words <math>x^g = g^{-1}xg</math>. Thus, we have <math>(x^g)^h = x^{gh}</math>.
+The latter notation is typically used in the theory of finite groups, when doing calculations involving conjugates and commutators. The primary advantage of this is that x^{gh} = (x^g)^h</math>.