Let G be a finite group of odd order and let F be a finite field. Suppose that V is an F G-module which carries a G-invariant nondegenerate 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.

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We obtain an explicit description of the Wells map for the automorphism group of a group extension in the full generality and investigate the dependency of this map on group extensions. Some applications are given.Crown