Structure is Group iff Semigroup and Quasigroup

Theorem

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

Then:

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

if and only if

$\struct {S, \circ}$ 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 $7.14$