Quadratic Forms on Finite Groups II

In my paper [1] I showed how a quadratic form on a finitely generated abelian group H led to one on a finite group G, and similarly for symmetric bilinear forms. The prototype for this is the relation between the intersection form on Hk(M ) for a manifold M, and the linking form on Hk.i(dM). I also showed that any form on G could so arise, but did not discuss uniqueness. Similar forms had already been considered by various authors [2; 3; 4] (including many of the results of [1] and some not to be found there); in particular, Kneser and Puppe [5] claimed that the symmetric bilinear form on G determined that on H up to stable equivalence, and proved this in the case \G\ odd. Complete proofs have since been given by Wilkens [Ph.D. thesis, University of Liverpool, 1971] and Durfee [Ph.D. thesis, Cornell University, 1971]; the former by lengthy matrix manipulations, the latter using a delicate p-adic analysis. Durfee in fact obtains the corresponding result for quadratic forms. The object of this paper is to present a direct and simple proof of the latter result, which arose out of work on [6; Chapter 8]. The argument can be generalised to replace Z by any order in a finite algebra over Q with anti-involution, without essential change.