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.

$\blacksquare$