I'm taking a course in abstract algebra, at the moment we are working on the Wedderburn theorem and I'm still unable to understand how to use it properly to solve exercises. I tried to solve a problem and explain my reasoning to see where are the holes in my understanding and to see if there's some other ways of thinking about it or suggestions for these type of problems.
The problem I tried to solve is the following:
Find the simple submodules of $\mathbb{C[Q_8]}$ (up to isomorphism).
First by Maschke's theorem I know that the group algebra is semisimple because char($\mathbb{C}$) doesn't divide char($\mathbb{Q_8}$). This means I can now use Wedderburn's theorem as it's a finite dimensional algebra. By this theorem I have the following decomposition: $$ \mathbb{C[Q_8]} \simeq \prod_{i \in I} M_{n_i}(\mathbb{C})$$ As it's an isomorphism, they both must have the same dimensions over $\mathbb{C}$. The dimension of $\mathbb{C[Q_8]}$ over $\mathbb{C}$ is 8. This implies that the only possible dimensions of the matrices algebras are $n_1=2,n_2=1,n_3=1,n_4=1,n_5=1$ because otherwise if $n_i = 1$ for $i \in \{ 1,...,8\}$ this would imply $\mathbb{C[Q_8]}$ is commutative which would imply $\mathbb{Q_8}$ is abelian as a group. The other cases $n_1 = 2, n_2 = 2$ cannot happen as there is always a submodule of a group algebra of dimension 1 (the one generated by $\sum_{h \in \mathbb{Q_8}} h$, is this correct?).
To find these subspaces isomorphic to $\mathbb{C}$ I considered the $x \in \mathbb{C[Q_8]}$ such that $i*x, j*x, k*x$ is a complex multiple of $x$. From these I found the four subspaces, one of them being the one that's guaranteed to exist. My problem is I can't seem to find the subspace isomorphic to $M_2(\mathbb{C})$. Any tips and help would be greatly apreciated.
I tried to solve the problem in another way. I read on the internet but couldn't prove it that $dim_{\mathbb{C}} Z(\mathbb{C[Q_8]}) = \# \text {conjugacy classes of $\mathbb{Q_8}$}$. I haven't worked that much with conjugacy classes but I gave it a try in this problem as I have seen that also, $dim_{\mathbb{C}} Z(\mathbb{C[Q_8]}) = \# \text {simples in the Wedderburn decomposition }$. By this I understood the number of matrices algebras in the product that is isomorphic to the algebra $\mathbb{C[Q_8]}$. In this particular case, the conjugacy classes I found are 5: the conjugacy class of $\{1\}$ and $\{-1\}$ as they both are in the center of the group. For each $i,j,k$ I found they are on their own conjugacy class. So indeed, the number of conjugacy classes is the same as the number of matrices algebras in the product, i.e 5. One question I have is if from these conjugacy classes I can get more information about these matrices algebras or they just determine the number of them in the product?
I would very happy if someone could point me to notes or books that explain these theorem and if possible with exercises similar to the one I asked.
Thanks for your help.