Representation Definition
Definition 2.1 ‒ Representation
A representation $D(G)$ of some group $G$ is a set of matrices, forming a group under matrix multiplication, for which there exists a surjective homomorphism $\mu: G \to D(G)$.
In other words, we associate each element in $G$ to an $n\times n$ matrix, such that the matrices respect the group multiplication
$$ \forall\quad g_{a}, g_{b} \in G,\quad D(g_{a}) D(g_{b}) = D(g_{a} g_{b}) $$Notice how this transformation need not be an isomorphism. However, if it is, the representation is called faithful.
Our interest in representations is obvious in the context of quantum mechanics. The states of the systems are vectors in a vector space and the operators acting on those states are matrices. So, these matrices form a representation of the group of symetries for the system.