The simplest proof of Burnside's theorem on matrix algebras (Q1826832)

By using induction on the dimension \(n\) of a vector space \(V\) (over an algebraically closed field and for \(n >1\)), it is shown that the only transitive or, equivalently, irreducible subalgebra \({\mathcal A}\) of linear transformations on \(V\) is the entire \({\mathcal L}(V)\), the algebra of all linear transformations on \(V\) (Burnside's theorem).