The Stability Classification for Abelian-by- Finite Groups and Modules over a Group Ring

Let G be a finite group. We consider the question: what are the superstable and co-stable abelian-by-G groups? (We consider these as structures in a suitable first order language, see § 1). We see that it is equivalent to ask: what are the superstable and co-stable Z[G]-modules? If we wish our answer to be such as to provide, at least in principle, some sort of uniform listing of all the (say) co-stable Z[G]-modules, then we immediately run into the problem of wild representation type. That is, except for the most simple groups G, even the finite (hence co-stable!) Z[G]-modules cannot be classified (given that 'wild implies unclassifiable'—cf. [22] for instance). Therefore, we restrict our question to those abelian-by-G groups, or Z[G]modules, where the underlying abelian group is required to come from some restricted class. In fact, most of our effort is devoted to the case where we have some finite bound on the exponent of the underlying abelian group. Roughly, we proceed as follows. We discuss the relationship between the classification problem for abelian-by-G groups and Z[G]-modules in general. Then we specialise to the case that the underlying abelian groups have exponent bounded by some fixed integer. We observe that the classification problem reduces to the prime-power case. In the prime-power case, we establish the boundary between the tame ('possibly classifiable') and wild (' unclassifiable') cases. This is not new, but each time that we prove wildness, at the same time we show undecidability of the theory of modules under consideration, thus lending support to the conjecture that 'wild implies undecidable' [25, Chapter 17]. In each tame case, we outline what is known about the co-stable and superstable modules. We also discuss briefly the problem of lifting classifiability of the Z[G]modules (when we have it) to classifiability of the corresponding abelian-by-G groups. We use the notation Cm for the multiplicative cyclic group of order m and Z/n for the additive cyclic group of order n.