Abstract Algebra: Theory and Applications · Problem 9.4.44

Normal Subgroups and Factor Groups — Thomas Judson

Mathematics — Abstract Algebra · regular-representation

Problem

Let

G

be a group and

g \in G

. Define maps

\lambda_g : G \to G

and

\rho_g : G \to G

\lambda_g(x) = gx

and

\rho_g(x) = xg^{-1}

. Show that

i_g = \rho_g \circ \lambda_g

is an automorphism of

G

. The isomorphism

g \mapsto \rho_g

is called the right regular representation of

G