 Monomiality of irreducible representations of finite p-groups

Every complex irreducible representation of a finite p-group is induced from a one-dimensional representation of a subgroup. Induct on the group order. Nonfaithful representations reduce to a smaller quotient. For a faithful representation of a nonabelian group, choose an abelian normal subgroup not contained in the centre. Restriction cannot be isotypic, since it would make that subgroup scalar and faithfulness would then force it central. The Clifford correspondence induces the representation from a proper subgroup; induction there and transitivity of induction complete the proof. Abelian groups have only one-dimensional complex irreducible representations.