Past exam of the groups and representations course of the University of Oxford / 2018 / 1 / b / Solution

The additive cyclic group ${\mathbb{Z}}_n$ is generated by $1$, with $n\cdot 1=0$. Every complex irreducible representation is one-dimensional by a. If $R(1)=z$, the defining relation requires $z^n=1$. Conversely each such root of unity defines a group homomorphism. Hence the complete list isChanging the representative of $g$ changes the exponent by an integer multiple of $2\pi i$, so the formula is well-defined. These representations are pairwise inequivalent because their values on the generator differ. The case $k=0$ is the trivial representation.