Vol. 78, No. 2, 1978

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Invariants of integral representations

Irving Reiner

Vol. 78 (1978), No. 2, 467–501

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 (mod p).

Mathematical Subject Classification 2000
Primary: 20C10
Received: 4 January 1977
Revised: 10 November 1977
Published: 1 October 1978
Irving Reiner