A group [representation](Representation.md) is a [homomorphism](Homomorphism.md) that maps a [group](Groups.md) onto a [vector space](Vector%20spaces.md). # Representation of a Finite Group A representation of a _finite group_ $\mbox{G}$ is the following homomorphism: $\Pi: \mbox{G}\rightarrow \mbox{GL}(\mathcal{V})$ where $\mathcal{V}$ is a complex vector space and $\mbox{GL}(\mathcal{V})$ is [[GL(n;F)]]. We can consider a representation to be an _action on a vector space by the group elements_ and thus we often write it as $g\cdot\mathcal{V}$ where $g\in\mbox{G}$. # Representations of Infinite Groups # sub-representations #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory #MathematicalFoundations/Algebra/AbstractAlgebra/RepresentationTheory