A finiteness lemma, Brauer's theorem and other irreducibility results

A technical lemma is proved for certain semigroups of matrices. It has seve ral applications to problems concerning irreducible semigroups satisfying s pectral conditions, e.g., submultiplicativity of spectrum. It is also used to give extensions of the following theorem of Brauer's: If U is a finite g roup of complex matrices, so that for some integer m, every U in U satisfie s U-m = I, then U has a representation over the cyclotomic field Q(w), wher e w is a primitive m-th root of unity.