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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.

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    $\begingroup$ 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. $\endgroup$
    – Pedro ♦
    Apr 16, 2017 at 18:58
  • $\begingroup$ 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. $\endgroup$
    – mathreadler
    Apr 16, 2017 at 19:24
  • $\begingroup$ @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. $\endgroup$
    – Matt Samuel
    Apr 16, 2017 at 19:39
  • $\begingroup$ @MattSamuel I did not mention extensive work but I do see your point. $\endgroup$
    – mathreadler
    Apr 16, 2017 at 19:47

1 Answer 1

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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:

Complex finite dimensional irreducible representation of abelian group

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.

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