Composition of Left Regular Representations

Theorem

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

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

Let $\lambda_x \circ \lambda_y$ be defined as the composition of the mappings $\lambda_x$ and $\lambda_y$.

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

$\lambda_x \circ \lambda_y = \lambda_{x * y}$

Proof

Let $z \in S$.

\(\ds \map {\lambda_x \circ \lambda_y} z\)

\(=\)

\(\ds \map {\lambda_x} {\map {\lambda_y} z}\)

Definition of Composition of Mappings

\(\ds \)

\(=\)

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

Definition of Left Regular Representation

\(\ds \)

\(=\)

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

Definition of Left Regular Representation

\(\ds \)

\(=\)

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

Semigroup Axiom $\text S 1$: Associativity

\(\ds \)

\(=\)

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

Definition of Left Regular Representation

$\blacksquare$

Sources

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