Set Equivalence of Regular Representations

Theorem

If $S$ is a finite subset of a group $G$, then:

$\card {a \circ S} = \card S = \left|{S \circ a}\right|$

That is, $a \circ S$, $S$ and $S \circ a$ are equivalent: $a \circ S \sim S \sim S \circ a$.

Proof

Follows immediately from the fact that both the left and right regular representation are permutations, and therefore bijections.

$\blacksquare$