 Such maximal tori "T" have a controlling role to play in theory of connected "G".Toroidal groups are examples of protori, which (like tori) are compact connected abelian groups, which are not required to be manifolds.Automorphisms of "T" are easily constructed from automorphisms of the lattice Z"n", which are classified by invertible integral matrices of size "n" with an integral inverse; these are just the integral matrices with determinant ±1.