= Group representation
{title2=$\rho:G\to\operatorname{GL}(V)$}
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= Representation
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A group representation is a <group homomorphism> from a <group> to the <general linear group> of a <vector space>. It realizes abstract group elements as invertible linear transformations.
The defining identities are $\rho(e)=I$ and $\rho(gh)=\rho(g)\rho(h)$, so an <inverse element> is represented by a <matrix inverse>. A choice of <basis> gives a <matrix representation>, but the underlying action does not depend on that choice. The action need not distinguish every group element: the <trivial representation> sends all of them to $I$.
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