Describe all the one dimensional complex representation of the cyclic group $C_n$ . Which ones are inequivalent?
attempt: Any 1 dimensional representation is irreducible. Every complex representation of $C_n$ can be expressed as a direct sum of irreducible reprentations. And by the fundamental theorem of finite abelian groups, $C_n$ is isomorphic to a direct sum of cyclic groups.
Can someone please help me? I dont really know what I have to show for the inequivalent part. Any feedback would be really helpful.
Thanks
The irreducible complex representations of $C_n$ are exactly the $1$-dimensional. Each one of them is specified by the requirement that a generator $c\in C_n$ acts (upon the elements of the $1d$ complex vector space) through multiplication by an $n$-th root of unity. Thus, if $\rho:C_n\rightarrow Aut V$ is the representation, then $$\rho(c)\cdot v=e^{2\pi i\frac{k}{n}}v, \ \ \ k=1,2,...,n$$ for a generator $c\in C_n$ and for any $v\in V$. In this way, each $n$-th root of unity provides an irreducible, complex representation of $C_n$ on $V$. These representations are mutually inequivalent. Thus, there are exactly $n$ inequivalent, irreducible, complex representations.