# Does $G/ker(\rho)$ being abelian tell us anything special about the representation $\rho$?

Suppose $G$ is a finite group and $\rho$ is a representation such that $\rho : G \rightarrow GL(V)$. Suppose that $G/ker(\rho)$ is Abelian. What does this tell us about $\rho$? In particular, how can we reach the conclusion that $\rho$ is the direct product of one-dimensional representations of $G$? I feel Maschke's theorem would be useful here, but I need some guidance.

This is an assignment question, so I am mainly looking for hints not full solutions. I want to figure it out on my own with hints and I hope to delete the question once I get an idea.

• I hope to delete the question once I get an idea. This is improper etiquette here: your posts are meant to help the community, and not only you. Please do not do this now, or ever.
– Pedro
Apr 16, 2017 at 18:58
• However it is not necessarily improper etiquette to ask for hints instead of full solutions. We do want to encourage both doing and showing own work on problems. Apr 16, 2017 at 19:24
• @mathreadler Whether askers should show extensive work is not universally agreed upon. Really it's used as a barometer to decide if the asker is trying to get free homework solutions, which some believe is somehow unjustified, even though copying down a solution you don't understand is more likely to hurt you than help. Apr 16, 2017 at 19:39
• @MattSamuel I did not mention extensive work but I do see your point. Apr 16, 2017 at 19:47

I assume $V$ is a complex vector space. This implies that the image of $\rho$ is commutative. You may use results about representations of finite commutative groups:
Since a representation of a finite group is the sum of irreducible representations, you deduce that $\rho$ is the sum of irreducible $1$-dimensional representations.