Representing Groups as matrices How to represent any group as group of matrices ?
Like how to represent dihedral (4) group (order $8$) as group of $2$ by $2$ matrices ?
How to represent direct product of $Z_2$ and $Z_2$ as a group of $4$ by $4$ matrices ?
similarly with quaternion group.
Is there any general procedure ?
 A: Any finite group $G$ can be represented as a group of matrices in the following way. First, realize $G$ as a subgroup of a permutation group acting on $\{1,\cdots,n\}$ (this is possible by Cayley's theorem). 
Then each $g \in G$ permutes the set $\{1,2,\cdots,n\}$, and we can define a map $G \to M_{n}(\mathbb R)$ from $G$ to the set of $n \times n$-matrices over $\mathbb {R}$: the permutation $g$ is sent to the matrix with columns $e_{g(i)}$, where $e_i$ are the basis vectors. 
One can check that this is actually a group homomorphism.
EXAMPLE Let $G=\mathbb Z_4$. Then $G$ can be realized as the permutation subgroup generated by the permutation $(1234)$. So we send $1 \in \mathbb Z_4$ to the matrix $$\begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0\end{bmatrix}.$$
And so on.
This is by no means the most "efficient" way of embedding $G$ in $GL_k$ (the group of invertible matrices). To do this efficiently, is one of the things representation theory deals with.
