Group Representation – Exercise 1

Suppose that ρ is a representation of G of degree 1. Prove that G\Ker(ρ) is abelian.

Solution:

For $$g,h\in G$$, $$\rho(ghg^{-1}h^{-1})=\rho(g)\rho(h)\phi(g^{-1})\rho(h^{-1}) (*)$$.

Since ρ is of degree 1, $$\rho(x)\in\mathbb{F} $$(field), so

\[(*)=\rho(g)\rho(g^{-1})\rho(h)\rho(h^{-1})=\]

\[\rho(gg^{-1})\rho(h^{1}h)=1\cdot 1 = 1.\]

We conclude that $$ghg^{-1}h^{-1}\in ker(\rho)$$, therefore

\[ghg^{-1}h^{-1} Ker(\rho) = Ker(\rho) \Longrightarrow (gh)Ker(\rho)= (hg)Ker(\rho).\]