# at.algebraic topology – What are the stable cohomology classes of the “orthogonal groups” of finite abelian groups?

Let $$A$$ be a finite abelian group, and equip it with a nondegenerate symmetric bilinear form $$langle,rangle : A times A to mathrm{U}(1)$$. Then you can reasonably talk about the “orthogonal group” $$O(A,langle,rangle)$$, i.e. the automorphisms of $$A$$ which preserve $$langle,rangle$$.

Actually, when $$A$$ has even order, the most standard version of “orthogonal group” is to choose a quadratic refinement $$q : A to mathrm{U}(1)$$ of $$langle,rangle$$, i.e. a quadratic function such that $$q(a_1 a_2) / q(a_1)q(q_2) = langle a_1,a_2rangle$$, and to ask for the group $$O(A,q)$$ of automorphisms preserving $$q$$. (When $$A$$ has odd order, there is a unique quadratic refinement, and so the two notions agree. In general, the quadratic refinements form a torsor for $$hom(A, mathbb{Z}/2)$$.) I can ask my question for both versions of $$O(A)$$, but in fact the one I care about is the “nonstandard” $$O(A,langle,rangle)$$ when $$A$$ has order a power of $$2$$.

I am interested in understanding the low $$mathbb{Z}/2$$-cohomology of $$O(A,langle,rangle)$$.

I specifically want to know about classes which are “stable”, but I am having trouble saying precisely what I want “stable” to mean. Certainly if I have two groups $$A, A’$$ each equipped with nondegenerate symmetric bilinear forms, then I only care about the image of restriction along $$O(A) subset O(A) times O(A’) subset O(A times A’)$$. But I expect that if there is a stronger notion of “stability”, then I only care about those classes.