Self-dual modules in characteristic two and normal subgroups

Abstract

We prove Clifford theoretic results which only hold in characteristic 2.Let G be a finite group, let N be a normal subgroup of G and let φ be an irreducible 2-Brauer character of N. We show that φ occurs with odd multiplicity in the restriction of some self-dual irreducible Brauer character θ of G if and only if φ is G-conjugate to its dual. Moreover, if φ is self-dual then θ is unique and the multiplicity is 1.Next suppose that θ is a self-dual irreducible 2-Brauer character of G which is not of quadratic type. We prove that the restriction of θ to N is a sum of distinct self-dual irreducible Brauer character of N, none of which have quadratic type. Moreover, G has no self-dual irreducible 2-Brauer character of non-quadratic type if and only if N and G/N satisfy the same property.Finally, suppose that b is a real 2-block of N. We show that there is a unique real 2-block of G covering b which is weakly regular with respect to N.