# Conjugation Classes in Group theory [duplicate]

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One reason for the importance of conjugacy classes is that if you fix the conjugating element, say $g \in G$, then the function $$i_g : G \to G, \quad i(h)=ghg^{-1}$$ is an automorphism of $G$. From this you deduce, for example, that the group theoretic properties of $h$ and of $ghg^{-1}$ are very similar, for example: the order of $h$ equals the order of $ghg^{-1}$; the centralizer of $h$ is isomorphic to the centralizer of $ghg^{-1}$; the normalizer of the cyclic subgroup $\langle h \rangle$ is isomorphic to the normalizer of the cyclic subgroups $\langle ghg^{-1} \rangle$; if I'm given a homomorphism $f : G \to A$ from $G$ to an abelian group $A$, $h$ is in the kernel of $f$ if and only if $ghg^{-1}$ is in the kernel of $f$. One can go on and on like this. It becomes a useful way to analyze an individual group, to classify groups, and so on.