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If $A$ is a finite subgroup of $G$ and $C_G(A):=\{g \in A:ag=ga\}$ is the centralizer of $A$, prove that $|G/C_G(A)|\leq(|A|-1)!$

If we look at the map $g \mapsto (a \mapsto gag^{-1})$, the image of this map has $|A|!$ elements (?)

How can I continue from here?

Thanks!

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    $\begingroup$ What is $C$? ${}$ $\endgroup$
    – Shaun
    May 15 at 14:16
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    $\begingroup$ The image does not have $|A|!$ elements. It maps into $S_n$, sure, but not surjectively. The map permutes the nontrivial elements of $A$. I'm sure this is a duplicate, but it may be hard to locate. $\endgroup$ May 15 at 14:17
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    $\begingroup$ If you mean $C=G$ this cannot be true: take $G=S_3$ and $A=\{(1),(12)\}$. Then $C_G(A)=A$, $|G:C_G(A)|=3$ and $(|A|-1)!=1$ ... $\endgroup$ May 15 at 14:40
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    $\begingroup$ $C$ is supposed to be the normalizer, surely. $\endgroup$ May 15 at 15:44
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    $\begingroup$ It's not "the definition of centralizer". Your equation has a $C$ (not $C_G(A)$, just $C$), which is undefined. $\endgroup$ May 15 at 15:45

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