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 $7.14$