Decidability for ℤ2 G‐lattices when G Extends the Noncyclic Group of Order 4

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.