Isotype subgroups of totally projective groups

Isotype knice subgroups of global Warfield groups

“…In fact, we shall establish such conditions in terms of an Axiom 3 characterization of the quotient group G/H. As a result, our Main Theorem (Theorem 4.9) will have roughly the same form as the characterization of the simply presented isotype subgroups of totally projective p-groups established in [6]. The principal result in that paper specializes to show that an isotype nice subgroup H of a totally projective p-group G is itself totally projective if and only if G/H has an Axiom 3 system of "separable" subgroups.…”

“…For example, m(h + x) mh + x implies that m(h + x) mx − x . When there is no danger of confusion with previous uses of the notation (see [6] or [3]), we shall occasionally write H N to indicate that H and N are almost strongly compatible. For later reference we remark that all of these compatibility relations are inductive.…”

“…If a subgroup H of G is p-separable for all primes p, then we say that H is locally separable in G. P. Hill proves in [6] that a simply presented isotype subgroup H of a torsion group G is necessarily locally separable in G. On the other hand, for a local Warfield group H to appear as an isotype subgroup of a local group G, H must be strongly separable in G (see [10]); in other words, to each g ∈ G there corresponds a countable subset {h n } n<ω ⊆ H such that if h ∈ H, g + h g + h n for some n < ω. But for global groups, it is possible for an isotype subgroup to be a Warfield group without being strongly separable in the containing group (see [9]).…”

“…In the application of Axiom 3 characterizations, there is an interplay between separability and compatibility that will be familiar to those who have studied either of the fundamental papers [6] or [3]. Such readers may indeed anticipate the following proposition (obtained by combining Propositions 4.5 and 4.6 of [12]).…”

“…We illustrate this phenomenon in our next proof. Although the following proposition is known in the case of p-primary groups (see [3] or [6]), we give a detailed demonstration which will serve as a model for our subsequent proofs. See [4,Theorem 5.1] for the genesis of this technique of applying the closed set method.…”