Composition of Right Regular Representations

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 $z \in S$.

\(\ds \map {\paren {\rho_x \circ \rho_y} } z\)

\(=\)

\(\ds \map {\rho_x} {\map {\rho_y} z}\)

Definition of Composition of Mappings

\(\ds \)

\(=\)

\(\ds \map {\rho_x} {z * y}\)

Definition of Right Regular Representation

\(\ds \)

\(=\)

\(\ds \paren {z * y} * x\)

Definition of Right Regular Representation

\(\ds \)

\(=\)

\(\ds z * \paren {y * x}\)

Semigroup Axiom $\text S 1$: Associativity

\(\ds \)

\(=\)

\(\ds \map {\rho_{y * x} } z\)

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 $2$