Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 19I / c / i / Solution

Yes. If $g$ generates the cyclic group $C_n$, define the one-dimensional complex representationIt is irreducible because it is one-dimensional, and it is faithful because the displayed scalar has multiplicative order $n$. Thus every $C_n$, including the trivial case $n=1$, has a faithful irreducible representation.