The decision problem for \(\mathbf Z C(p^3)\)-lattices with \(p\) prime (Q1128176)

The author studies undecidability for lattices over a group ring \({\mathbb Z}G\) for \(G\) a finite group. In particular, \(G\) is a finite group such that for some prime \(p\) a Sylow \(p\)-subgroup \(S\) of \(G\) is isomorphic to \(C(p^3)\). For \(p\) an odd prime the theory of \({\mathbb Z}G\)-lattices is undecidable. The proof involves showing via an interpretation that the theory of \({\mathbb Z} _p S\)-lattices is undecidable and then similarly the theory of \({\mathbb Z}_pG\)-lattices is undecidable. The author considers the decision problem where \(G\) is cyclic of order 8. The implications of a positive solution to this particular decision problem are discussed.