Irreducible representations of $\mathbb{Z}_{8}$ Let $G = \mathbb{Z}_{8}$. How do I determine the (isomorphism classes) irreducible representations of $G$ over $\mathbb{C}$? I do not know how to approach such a problem, and if someone can give a general approach to finding the irreducible representations of a finite $G$ over $\mathbb{C}$, I would appreciate it. 
 A: There are entire books written about finding irreducible
representations of finite groups  over the complex field.
You should consult these for general details. It is known that all 
irreducible representations of a finite abelian group have degree $1$.
More specially, the irreducible representations of a finite cyclic 
group are all powers of a linear representation generator as follows.
Suppose that $\,n>0\,$ is an integer, a multiplicative group 
$\,G \cong \mathbb{Z}_{n}\,$ with generator $\,g\in G,\,$ and 
$\,\zeta:=e^{2\pi i/n}.\,$  The linear mappings
$\, \rho_k(g^j): z\to \zeta^{kj}z\,$ (where $\,z\in\mathbb{C}\,$
and $\,\mathbb{C}\,$ is regarded as a one dimensional complex vector
space over itself) for all integer $\,0 \le k < n\,$
are all of the irreducible representations of $\,G.\,$ These are
all degree one linear maps and hence $\,\chi_k(g^j) := \zeta^{kj}\,$
are also all the irreducible characters.
A: Since $\mathbb Z_8$ is a finite abelian group, every irreducible finite-dimensional representation of $\mathbb ZZ_8$ is actually $1$-dimensional and it is given by an action of $\mathbb Z_8$ on $\mathbb C$ given by $g.v=\varphi(g)v$, where $\varphi\colon\mathbb Z_8\longrightarrow\mathbb C\setminus\{0\}$ is a group homomorphism.
A: My Maschke's theorem, $\mathbb C[G]$ is a semisimple ring for any finite group, that is, it is a finite product of matrix rings over $\mathbb C$.  (Maschke's theorem is more general, but the claim that the matrices are all over $\mathbb C$ depends on the algebraic closedness of $\mathbb C$.)
If additionally $G$ is abelian, then you have to have that all the matrix rings are trivial, so that $\mathbb C[G]\cong \mathbb C^{|G|}$.
In a finite product of fields, you can exhibit all isoclasses of simple modules by listing the ideals you get by selecting an index $i$ in the product and zeroing out entries outside of that index.
That covers your case.  The case of nonabelian $G$, you'd have to consider the simple submodules of matrix rings: however each one of them only contributes a single simple submodule to the product, so it is not much harder.
