Proposition 9.1.13.label Let $G$ be a group, $Y$ be a set, and $x, y \in G$, then:
- (1)
$L_{xy}= L_{x}L_{y}$.
- (2)
$R_{xy}= R_{x}R_{y}$.
Proof. (1): For any $f \in Y^{G}$ and $z \in G$,
\[L_{xy}f(z) = f((xy)^{-1}z) = f(y^{-1}x^{-1}z) = L_{y}f(x^{-1}z) = L_{x}L_{y}f(z)\]
$\square$
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