I just read the firct chapter of an introduction to representation theorey and tried solving some given questions, and im stuck with this one.
Let G $G=\left< x \right> $ be a cyclic group of order n and V a CR-Modul (C denoting the Complexe Numbers) with finite dimension. Give a simple elementary proof that V is a direct sum of Submodules ${ V }_{ r }$ $1\le r\le n$, with the propertie that x acts on $${ V }_{ r }$$ as multiplication with ${ e }^{ \frac { 2\pi r }{ n } }$
Im really new to the topic so im not sure where to start. Since it is asked for a simple proof i thought about induction. But on what should i do the induction, dimension of ${ V }_{ r }$ or the number r of submodules in the direct sum decomposition. Dimension doesn't make this much sense to me because i think every sumbmodule in the decomposition should have dimension 1. And for the second part one should consider the associated group action to the KG- Module however i dont understand why G being cyclic limits the KG-Module to the unit circle and not any other numbers in the complexe plane. Is this approach helpfull or is it completley off. Could you help me out with some instructions