Some remarks on Clifford's theorem and the Schur index

Abstract

Some time ago Clifford described the behavior of an irreducible representation of a finite group when it is restricted to a normal subgroup. One interesting case in this description requires that the representation be written in an algebraically closed field. In this note we shall consider this case when the field is “small”. We describe conditions under which an irreducible representation decomposes as the tensor product of two projective representations. Our approach uses certain subalgebras of the group algebra and the course of the discussion makes it fairly easy to keep track of the division algebras that appear. Hence we obtain some information about the Schur index. We apply this information to the case where the group is a semi-direct product PA of a p-group P and a normal cyclic group A. If is an algebraic number field and χ an absolutely irreducible character of PA, then there normal subgroups P1 ⊇ P2 ⊇ P3 of P which contain CP(A) such that the Schur index of mg(χ) of χ over F divides 2[P1: P2]e where e is the exponent of P2/P3. The factor 2 can be omitted if p ≠ 2. Some conditions are available to restrict the Pi further. © 1970 by Pacific Journal of Mathematics.