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Direct sum of subspaces

Source: Miklós Schweitzer 2014, problem 6

December 23, 2014
vectorabstract algebrasuperior algebrasuperior algebra unsolved

Problem Statement

Let ρ:G→GL(V)\rho:G\to GL(V) be a representation of a finite pp-group GG over a field of characteristic pp. Prove that if the restriction of the linear map ∑g∈Gρ(g)\sum_{g\in G} \rho(g) to a finite dimensional subspace WW of VV is injective, then the subspace spanned by the subspaces ρ(g)W\rho(g)W (g∈G)(g\in G) is the direct sum of these subspaces.
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