Find the character table of $U_{16}$.
Could you give me a hint or a start? Thank you.
Find the character table of $U_{16}$.
Could you give me a hint or a start? Thank you.
The number of irreducible representations of a group is the number of conjugacy classes of that group. In an abelian group each element is its own conjugacy class, so there are $|G|$ irreducible representations for an abelian group $G$. Now, the number of degree one complex representations of a group $G$ is $[G:G']$, which of course is $|G|$ in an abelian group $G$. So all irreducible complex representations are degree one. Note that we could also have seen this from the formula $\sum_{i=1}^{|G|}d_i=|G|$ where $d_i$ denotes the degree of each irreducible representation.
So, your problem is reduced to finding all $16$ homomorphisms from $U_{16}\rightarrow \mathbb{C}^*$. I bet you can do this.