Convention:Conjugation

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 $$x^{gh} = (x^g)^h$$, 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.