Opposite Group of Opposite Group

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

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

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

Then:


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

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


 * $a \circ b = b * a = a \circ' b$

by definition of opposite group.

Hence the result.