Regular Representations in Group are Permutations

Theorem

Let $\left({G, \circ}\right)$ be a group.

Let $a \in G$ be any element of $G$.

Then the left regular representation $\lambda_a$ and the right regular representation $\rho_a$ are permutations of $G$.

Proof

This follows directly from the fact that all elements of a group are by definition invertible.

Therefore the result Regular Representations of Invertible Elements are Permutations applies.

$\blacksquare$

Sources

• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.6$: Example $18$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 28 \beta$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 35.8$