I'm trying to search the through the finite subgroups of $GL(n, Z)$. Since each is isomorphic to a subgroup of $GL(n, 3)$, I identify its subgroups, and then I want to lift them to $GL(n, Z)$. Or exclude them by proving they cannot be lifted.
Once I have the generators of a subgroup of $G(n, 3)$, I can lift the generators successively: modulo $3^2$ so that the relators are consistent, then modulo $3^3$ etc until multiplication modulo $3^k$ gives the same result as standard multiplication.
How do I deal with termination: will this always terminate or can it fail? Also, is there any other way of doing the lifting?
The subgroups I'm interested are groups of rotations that leave a lattice invariant.