Existence Theorems for Warfield Groups

Abstract

Abstract. Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.

Realization Theorems for Valuated $p^n$-Socles

“…Another definition of the Ulm function of a general valuated group is given in [10], and it is easy to verify that these agree for valuated p n -socles. In general, when we say f : y 3 g is a function, we mean that its support is contained in some ordinal d, and we identify f with its restriction f : d 3 g.…”

The fully invariant subgroups of local Warfield groups

“…Let C = i∈ω B i and C = i∈ω B i . The inclusions B i = B ⊆ B = B i induce an inclusion C ⊆ C, under which C is fully invariant in C. By [6,Corollary 7], C is simply presented. Because tC is simply presented, we conclude from Corollary 2 that C is a Warfield group with simply presented torsion.…”

A homological characterization of certain subgroups of Abelianp-groups

“…The above construct]on of My is a simple modification of the constructions of the Zp-modules Pp of [8] and Qa of [5]; M, can play, in the theory of Warfield groups developed in [9] and [4], the same role l~layed by the groups Pa in the theory oftotaUy projectivep-groups (see [8]) and by the groups Qp in the theory of balanced projective Zp-modules (or KT-modules) (see [5]). …”

“…The exact sequence (3) is sequentially pure if, for every increasing sequence of ordinals and ~'s _v=(~r,),~N, the following sequence is exact: (6) is exact for every v such that a,,<), for every nEN. The converse is not generally true, as the theory of Warfield groups and balanced projective groups shows ( [3], [4], [9]); but according to the fact that torsion Warfield groups are totally projective, the converse holds if all groups in (3) ; it follows that the Ulm indicator of b" is -~(a0, co, ~, ...)=>v. Suppose now that the claim is true for n-1 (n>l); then, if b+A has exponent n, pb+A has exponent n-1 and pb+AEC (al, a2, 9 .... a ..... ); we can suppose pbEp*lB with bEp~ for the second equality in (5) and for al-_>o'0+1; by induction hypothesis, there exists xEB(al, a2, ..., a~, .,.) ).-pure), then it is also balanced (resp.…”

“…p'A ~ p~B ~ p~C ~ 0 is exact; (3) is 2-balanced, where )~ is a limit ordinal, if(4) follows for every a<2, which is equivalent to the following properties:(5) p"BAA = p~A and (p"B+A)/A = p"C Va < 2.…”