Conjugacy class

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A conjugacy class is a particular subset of a group.

Let $G$ be a group. Consider the action of $G$ on itself by inner automorphisms. The orbits of $G$ are then called conjugacy classes. By expanding the definition, it is easy to show that two elements $g$ and $g'$ are in the same conjugacy class iff there is an element $x$ such that $g' = x^{-1}gx$.

Two subsets $H$ and $H'$ of $G$ are called conjugate if there exists $\alpha \in G$ for which $H$ is the image of $H'$ under $\text{Int}(\alpha)$.

The character of any group $G$ are constant on conjugacy classes.

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