Decidability for ℤ2G-lattices whenG Extends the Noncyclic Group of Order 4

Abstract

Let G be the direct sum of the noncyclic groupof order four and a cyclic groupwhoseorderisthe power pn of some prime p. We show that ℤ2 G-lattices have a decidable theory when the cyclotomic polynomia (x) is irreducible modulo 2ℤ for every j ≤ n. More generally we discuss the decision problem for ℤ2 G-lattices when G is a finite group whose Sylow 2-subgroups are isomorphic to the noncyclic group of order four.