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The center of a group $G$ is

$\{z|z \in G$ and for all $g \in G, gz = zg\}$

Ie...the set of elements that commute with every $g \in G$.

But $gz = zg$ can be written as $g^{-1}zg = z$

So it seems the center of $G$ can also be described as the set of all elements of $G$ that are invariant under conjugation with any element of $G$...is that correct?

If so, it seems that commute and conjugate are the same thing?

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You're right about your alternate definition of the center. We can say that $g$ and $z$ commute if and only if $z$ is invariant under conjugation by $g$, so "commute" and "conjugate" are related, but they aren't the same thing.

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    Thanks, that clears it up.2012-11-13