Self-dual modules for finite groups of odd order

Let G be a finite group of odd order and let F be a finite field. Suppose that V is an FG-module which carries a G-invariant non-degenerate bilinear form which is symmetric or symplectic. We show that V contains a self-perpendicular submodule if and only if the characteristic polynomials of some specified elements of G (regarded as linear transformations of V) are precisely squares. This result can be applied to the study of monomial characters if the form on V is symplectic, and self-dual group codes if the form is symmetric.