Group is Quasigroup

Theorem

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

Then $\struct {G, \circ}$ is a quasigroup.

Proof

By the definition of a quasigroup it must be shown that:

for all $g \in G$, the left and right regular representations of $\lambda_g$ and $\rho_g$ are permutations on $G$.

This follows immediately from Regular Representations in Group are Permutations.

$\blacksquare$