Constructing representations of the finite symplectic group Sp(4,q)

Let G be a finite group and chi be an irreducible character. We say that a subgroup H is a chi-subgroup if the restriction chi(H) of chi to H has at least one linear constituent of multiplicity 1. Not every pair (G, chi) has a chi-subgroup, but chi-subgroups can be found in many cases. The existence of such subgroups is of interest for several reasons, one being that knowledge of a chi -subgroup enables us to give a simple construction of a matrix representation of G affording chi. In this paper we show that, when G = Sp(4, q) where q is a power of an odd prime p and H is a Sylow p-subgroup of G, then H is a chi-subgroup for every irreducible character chi (with one exception). We also find a p-subgroup which is a chi-subgroup for the exceptional character. (c) 2005 Elsevier Inc. All rights reserved.