Opposite Group of Opposite Group

Theorem
Let $\struct {G, \circ}$ be a group.

Let $\struct {G, *}$ be the opposite group to $\struct {G, \circ}$.

Let $\struct {G, \circ'}$ be the opposite group to $\struct {G, *}$.

Then:


 * $\struct {G, \circ} = \struct {G, \circ'}$

Proof
We have, for all $a, b \in G$:


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

by definition of opposite group.

Hence the result.