Isotype Warfield Subgroups of Global Warfield Groups

Isotype knice subgroups of global Warfield groups

“…But of course, in the global setting, the separability condition will need to be suitably modified; and moreover, each subgroup in the Axiom 3 system for G/H will be required to be a k-subgroup, a new type of subgroup whose definition appears below. We should also mention that in the case of p-local Warfield groups in [10] and of global Warfield groups in [12], the isotype subgroups that are themselves Warfield were characterized. However, those results have the unpleasant feature that one must consider properties of the quotients (H + N )/N in G/N as N ranges over an entire Axiom 3 system of knice subgroups for G. In this section, we prove two theorems that will play pivotal roles in the sequel.…”

“…We begin by recalling the notions of compatibility used by us in [12]. First, if p is a prime, we say that the subgroups H and N of G are p-compatible in G if for each h ∈ H and x ∈ N there corresponds an x ∈ H ∩ N such that |h + x| p |h + x | p .…”

“…With this new terminology, [12,Theorem 2.1] shows that any knice subgroup of a k-group is a k-subgroup. …”

“…The details may be found in [7] and will not be reviewed here. The general facts we cite from [7], [8], [11] and [12] will be adequate for our purposes. Suffice it to say that the exact definitions of primitive element and * -valuated coproduct in a global group G can be formulated in terms of abstract height matrices and various associated fully invariant subgroups of G. We use the term global group for an arbitrary mixed abelian group to distinguish the groups we study from p-local groups, the simpler setting in which Warfield groups were first introduced.…”

“…(1), (2), and (3) in the preceding proposition are, respectively, Theorem 3.2, 3.3, and 3.7 in [7]; while (4) is Corollary 1.10 from [8] and (5) is [12,Corollary 2.3], whose proof uses both (4) and the characterization of knice subgroups given in Proposition 1.3 below.…”