Schur's lemma Irreducible representations of abelian groups are 1-dimensional Let be an Abelian group. As a corollary of Schur's lemma, if is a (complex) Irrep then it is a 1-dimensional irrep. #m/thm/rep Moreover, any abelian irrep is 1-dimensional. ProofIf is abelian, then so are all its irreps. Let be an abelian irrep of on , so for every , and therefore is a multiple of the identity for all , so every subspace of is invariant under . Thus must be 1-dimensional in order for to be an irrep. See also Main theorem. #state/tidy | #lang/en | #SemBr