Order of Subset Product with Singleton/Proof 1
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $X, Y \subseteq \struct {G, \circ}$ such that $X$ is a singleton:
- $X = \set x$
Then:
- $\order {X \circ Y} = \order Y = \order {Y \circ X}$
where $\order S$ is defined as the order of $S$.
Proof
From Regular Representations of Subset Product, we have that the left regular representation of $\struct {S, \circ}$ with respect to $a$ is:
- $\lambda_x \sqbrk S = \set x \circ S = x \circ S$
The result then follows directly from Regular Representation of Invertible Element is Permutation.
$\blacksquare$