Representation theory

The mathematical branch that connects groups and vector spaces is called representation theory. A representation of a finite group $G$ on a finite-dimensional vector space $V$ is a homomorphism $$\rho :G\to \text{GL}(V):g\mapsto \rho (g)$$ of the group $G$ to the general linear group $\text{GL}(V)$, such that every group element $g$ is associated to an element of the general linear group. In other words, we associate every group element $g$ with an $n\times n$-matrix $\rho (g)$. The term homomorphism means that group structure is preserved: $$\mathrm{\forall }{g}_1,g_2\in G:\rho (g_1\cdot g_2)=\rho (g_1)\rho (g_2),$$ which means that the matrix representation $\rho (g_1\cdot g_2)$ of the group multiplication of two group elements $g_1$ and $g_2$ is the matrix product of their respective matrix representations $\rho (g_1)$ and $\rho (g_2)$.