Center of Opposite Group

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $\left({G, *}\right)$ be the opposite group to $G$.

Let $Z \left({G, \circ}\right)$ and $Z \left({G, *}\right)$ be the centers of $\left({G, \circ}\right)$ and $\left({G, *}\right)$, respectively.

Then:


 * $Z \left({G, \circ}\right) = Z \left({G, *}\right)$

Proof
We have, for $g \in G$:

Hence the result, by definition of set equality.