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On characterization of monomial representations of discrete supersolvable groups

We prove that an abstract (possibly infinite dimensional) complex irreducible representation of a discrete supersolvable group is monomial if and only if it has finite weight. We also prove a general result that implies converse of Schur's lemma holds true for certain induced representations of finitely generated discrete groups. At last, we work out example of infinite dihedral group and prove that it is a monomial group.