Theorem. representations are Group-Algebra modules [fgap-001I]
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Representation.asModuleRepresentation.ofModuleRep.equivalenceModuleMonoidAlgebra
L∃∀N

    Let \(k\) be a field, \(G\) a group, and \(V\) a \(k\)-vector space. Giving a linear representation \(\rho :G\to \operatorname {GL}_k(V)\) is equivalent to giving a unital left \(k[G]\)-module structure on \(V\), up to the identity type synonyms used to keep the 2 scalar actions distinct. The action associated to \(\rho \) is \[ (\sum _g a_g[g])\cdot v=\sum _g a_g\rho (g)v. \] Conversely, restriction of a unital \(k[G]\)-action to the units \([g]\) recovers \(\rho \). Invariant subspaces are submodules and intertwining linear maps are \(k[G]\)-linear maps.

    Proof. 

      The extension in the Group Algebra and its universal extension supplies the algebra homomorphism \(k[G]\to \operatorname {End}_k(V)\). Associativity and the unit law give a module. In the other direction, multiplication by each \([g]\) is invertible, and the module law gives the group law. These operations are inverse. See [sengupta2010representations, sec. 3.2, pp. 40--41].