Nondegenerate symmetric bilinear forms on finite abelian $2$-groups

Abstract:

Let ${\mathcal {B}_2}$ be the semigroup of isomorphism classes of finite abelian $2$-groups with a nondegenerate symmetric bilinear form having values in $Q/{\mathbf {Z}}$. Generators for ${\mathcal {B}_2}$ were given by C. T. C. Wall and the known relations among these generators were proved to be complete by A. Kawauchi and S. Kojima. In this article we describe a normal form for such bilinear forms, expressed in terms of Wall’s generators, and as a by-product we obtain a simpler proof of the completeness of the known relations.