# Structure is Group iff Semigroup and Quasigroup

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## Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Then:

- $\struct {S, \circ}$ is a group

- $\struct {S, \circ}$ is both a semigroup and a quasigroup.

## Proof

### Sufficient Condition

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

Then a fortiori $\struct {S, \circ}$ is a semigroup.

From Regular Representations in Group are Permutations:

- for all $a \in S$, the left regular representation and the rightt regular representation are permutations of $S$.

Hence by definition $\struct {S, \circ}$ is a quasigroup.

$\Box$

### Necessary Condition

Let $\struct {S, \circ}$ be both a semigroup and a quasigroup.

By definition of quasigroup:

- $\forall a \in S$, the left and right regular representations $\lambda_a$ and $\rho_a$ are permutations on $S$.

It follows from Regular Representations in Semigroup are Permutations then Structure is Group that $\struct {S, \circ}$ is a group.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.14$