Conjugate (group theory)

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Let $G$ be a group operating on a set $S$. An element $y\in S$ conjugate to an element $x\in S$ if there exists an element $\alpha \in G$ such that $y = \alpha x$. The relation of conjugacy is an equivalence relation. The set of conjugates of an element $x$ of $S$ is called the orbit of $x$.

Note that this definition conforms to the notion of complex conjugate. Indeed, under the group of field automorphisms on the complexe numbers that do not change the reals, the orbit of a complex number $z$ is the set $\{z, \bar{z}\}$.

If $H$ is a subset of a group $G$, the conjugate of $H$ usually means the conjugate of $H$ under the group of inner automorphisms acting on the subsets of $G$. If $H$ is a subgroup of $G$, any conjugate of $H$ is also a subgroup, as for any $a \in G$, \[(aHa^{-1})(aHa^{-1}) = aHHa^{-1} = aHa^{-1}.\]

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