Convention:Conjugation

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:

$c_g:x\mapsto gx{g}^{-1}$

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 $x^{gh}=(x^g{)}^h$, which is convenient for results.

Because of the inherent left-right symmetry in groups, the main definitions remain the same whatever convention we choose. It is important to remember the conventions only when trying to follow a notation-heavy proof.