Let ZG be the integral group
ring of a finite group G. A ZG-lattice is a left G-module with a finite free Z-basis. In
order to classify ZG-lattices, one seeks a full set of isomorphism invariants of a
ZG-lattice M. Such invariants are obtained here for the special case where G is cyclic
of order p2, where p is prime. This yields a complete classification of the
integral representations of G. There are also several results on extensions
of lattices, which are of independent interest and apply to more general
situations.
Two ZG-lattices M and N are placed in the same genus if their p-adic
completions Mp and Np are ZpG-isomorphic. One first gives a full set of genus
invariants of a ZG-lattice. There is then the remaining problem, considerably more
difficult in this case, of finding additional invariants which distinguish the
isomorphism classes within a genus. Generally speaking, such additional invariants
are some sort of ideal classes. In the present case, these invariants will be a pair of
ideal classes in rings of cyclotomic integers, together with two new types of
invariants: an element in some factor group of the group of units of some finite ring,
and a quadratic residue character (modp).
