# Definition:Opposite Group

## Theorem

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

We define a new product $*$ on $G$ by:

- $\forall a, b \in G: a * b = b \circ a$

The algebraic structure $\struct {G, *}$ is called the **opposite group** to $G$.

## Also see

- Opposite Group is Group, demonstrating that this is indeed a group
- Definition:Group Antihomomorphism
- Definition:Opposite Ring

- Results about
**opposite groups**can be found here.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26 \epsilon$