A characterization of Warfield groups

Abstract

Abstract.Local Warfield groups are generalizations of totally projective p-groups. This paper presents a characterization of local Warfield groups which is the analogue of the description of totally projective p-groups as groups with a nice composition series.

Axiom 3 modules

“…What is perhaps unexpected though is that the equivalence of these two classes of Zp-modules is a byproduct of the proof of Theorem 3.6. We do, however, require a characterization, due to J. Moore [12], of Warfield modules which is proved fairly directly from the above definition. A submodule N of G is said to be quasi-sequentially nice provided it is nice in G and for each g E G there is a k < oj such that pkg + N contains an element g1 having the same height sequence as pkg + N. When N is nice in G, this latter condition is easily seen to be equivalent to N ffi (pkg + z) being a valuated coproduct for some z E N. Moore shows that a p-local group G is a Warfield module if and only if G is the union of a smooth chain {Na}a<p of quasi-sequentially nice submodules with Nq = 0 and Na+i/Na cyclic for each a.…”

“…As quasi-sequentially nice submodules have entered into the study of Warfield modules not only in [12] but also in [8], it seems appropriate to comment on how this concept compares with kniceness. That it is a strictly weaker notion is clear since 0 is always a quasi-sequentially nice submodule of G, but is not knice unless G is a fc-module.…”