# Composition of Right Regular Representations

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## Theorem

Let $\struct {S, *}$ be a semigroup.

Let $\rho_x$ be the right regular representation of $\struct {S, *}$ with respect to $x$.

Let $\rho_x \circ \rho_y$ be defined as the composition of the mappings $\rho_x$ and $\rho_y$.

Then $\forall x, y \in S$:

- $\rho_x \circ \rho_y = \rho_{y * x}$

## Proof

Let $z \in S$.

\(\displaystyle \map {\paren {\rho_x \circ \rho_y} } z\) | \(=\) | \(\displaystyle \map {\rho_x} {\map {\rho_y} z}\) | Definition of Composition of Mappings | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\rho_x} {z * y}\) | Definition of Right Regular Representation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \paren {z * y} * x\) | Definition of Right Regular Representation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle z * \paren {y * x}\) | Semigroup Axiom $S1$: Associativity | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\rho_{y * x} } z\) | Definition of Right Regular Representation |

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $2$