A technical lemma is proved for certain semigroups of matrices. It has several applications to problems concerning irreducible semigroups satisfying spectral conditions, e.g., submultiplicativity of spectrum. It is also used to give extensions of the following theorem of Brauer's. IfUis a finite group of complex matrices, so that for some integerm, everyUinU, satistiesUm=IthenUhas a representation over the cyclotomic fieldQ(ω), where ω is a primitivem-th root of unity.