# Extensions of orthogonal groups of torsion quadratic forms.

Hi. This is related to a question I asked earlier. The setup is:

Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = \langle e_i, e_j \rangle$, where $\langle x,y \rangle = q(x+y) - q(x) - q(y)$ is the associated bilinear form. Let $L^*$ be the dual lattice to $L$, and $G_L = L/L^*$, and assume $|G_L|$ is odd. $q$ then descends to a $\mathbb{Q}/\mathbb{Z}$-valued quadratic form $q_G$ on $G_L$. Let $O(G_L)$ be the group of automorphisms of $G_L$ that preserve $q_G$.

Since $O(G_L)$ acts on the finite abelian group $G_L$, it is natural to consider extensions of $O(G_L)$ by $G_L$. These correspond to $H^2(O(G_L), G_L)$. Have such extensions been considered before? Is there an algorithm or method to calculate them (given $K_{ij}$)? Any references would be appreciated.

Thanks!

(edit: corrected mistake above, thanks to B. Conrad for pointing it out)

-