Convention:Conjugation

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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, 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.