Let G be any finite group. Questions: 1) Is there always some n such that there is some n-dimensional lattice L whose isometry group (keeping the origin fixed) is isomorphic to G ? Call such an isometry group I_0(L). 2) If so, is there always some n and some lattice L that is a (not nec. proper) subgroup of the integer lattice Z^n, such that I_0(L) is isomorphic to G ? ------------------------------------------ Note: It's easy to see that if n = #(G), then the symmetric group S_n is a *subgroup* of I_0(Z^n), and hence, so is G. Close, but not quite what we are looking for. ------------------------------------------ (We define a n-dimensional lattice to be any discrete subgroup -- isomorphic to the group Z^n -- of the abelian group and Euclidean space R^m, necessarily for some m >= n.) --Dan