Set Equivalence of Regular Representations
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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.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 41.2$ Multiplication of subsets of a group