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 ℱ is an algebraic number field and χ
an absolutely irreducible character of PA, then there normal subgroups
P1⊇ P2⊇ Pa of P which contain CP(A) such that the Schur index mf−(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.
