On the involutions fixing the class of a lattice

With any integral lattice Λ in n-dimensional Euclidean space we associate an elementary abelian 2-group I(Λ) whose elements represent parts of the dual lattice that are similar to Λ. There are corresponding involutions on modular forms for which the theta series of Λis an eigenform; previous work has focused on this connection. In the present paper I(Λ) is considered as a quotient of some finite 2-subgroup of O_n(ℝ). We establish upper bounds, depending only on n, for the order of I(Λ), and we study the occurrence of similarities of specific types.