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What is the order of $x$ in a group of order 21 that contains a conjugacy class of order 3?

I know the answer is 7 because the size of the conjugacy class of $x$ equals the index of its centralizer: $[G:Z(x)]$. However, why is the order of the element equal to the cardinality of the centralizer?

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  • $\begingroup$ In general, it's not! (see answer below) $\endgroup$
    – Doc
    Dec 4, 2013 at 2:35

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Well, $\langle x\rangle$ is certainly a subgroup of its centralizer $Z(x)$ (which does have order 7) and a group of order $7$ can have only one proper subgroup. If $x=1$, then its class size would be $1$, because the centralizer of $1$ is the entire group.

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