NICE ELONGATIONS OF PRIMARY ABELIAN GROUPS

Suppose N is a nice subgroup of the primary abelian group G and A = G/N . The paper discusses various contexts in which G satisfying some property implies that A also satisﬁes the prop- erty, or visa versa, especially when N is countable. For example, if n is a positive integer, G has length not exceeding ω 1 and N is countable, then G is n -summable iﬀ A is n -summable. When A is separable and N is countable, we discuss the condition that any such G decomposes into the direct sum of a countable and a separable group, and we show that it is undecidable in ZFC whether this condition implies that A must be a direct sum of cyclics. We also relate these considerations to the study of nice bases for primary abelian groups.


Introduction and Terminology
By the term "group" we will mean an abelian p-group, where p is a prime fixed for the duration.Our group theoretic terminology and notation will generally follow [11].We will say a group G is Σ-cyclic if it is isomorphic to a direct sum of cyclic groups.We will also utilize the language of valuated groups and valuated vector spaces (see [22] and [12], respectively); for example, if Y is a valuated group, the letter v will be reserved for its valuation, and for any ordinal α, by Y (α) we will mean the subgroup {y ∈ Y : v(y) ≥ α}.We will implicitly assume that all valuated vector spaces are over Z p .
If 0 → X → G → A → 0 is a short exact sequence, we will routinely identify X with an actual subgroup of G and A with the quotient group G/X; we will say that G is an elongation of A by X.We then say that G is a nice-elongation of A if X is a nice subgroup (i.e., for every y ∈ G, the coset y + X has an element of maximum height).Further, if X is countable, then we will say G is an ℵ 0 -elongation of A by X.We can clearly combine these and speak of nice-ℵ 0 -elongations.
The main purpose of this paper is to investigate how certain properties of groups are preserved under nice-ℵ 0 -elongations, though we will occasionally prove results that extend beyond the countable case.For example, if λ ≤ ω 1 is an ordinal, then a group G is a C λ -group if for every α < λ, if H is a p α -high subgroup of G (i.e., H is maximal with respect to the property that H ∩p α G = {0}), then H is a dsc-group (i.e., a direct sum of countable groups).We verify that if G is a nice-ℵ 0 -elongation of A, then G is a C λ -group iff A is a C λ -group (Corollary 2.4).
In another direction, a valuated group Y will be said to be Honda if it is the ascending union of subgroups X m whose value spectra {v(x) : x ∈ X m } are finite (i.e., they are value-finite).Note that if Y is actually countable, then it is clearly Honda, since it is the ascending union of finite subgroups.
If G is a group with a subgroup Y , then restricting the height function on G to Y gives a valuation on Y .A group G is summable if G[p] is isometric to a free valuated vector space.More generally, following [8], if n is a positive integer, a reduced group G is n-summable if G[p n ] is isometric to a valuated direct sum of countable valuated groups, and G is n-Honda if G[p n ] is Honda as a valuated group.In [8] it was shown that if G is n-summable, then (a) it is summable, so that p ω1 G = {0} (see, for example, [11,Theorem 84.3]);(b) the valuated group G[p n ] is determined up to isometry by its Ulm function; (c) G is n-Honda iff it is n-summable and has countable length (i.e., p α G = {0} for some countable α).In fact, these results were proven in the category of valuated p n -socles (which extends the idea of a valuated vector space).Note that (c) generalizes the classical criterion (due to Honda) for the summability of a group of countable length (see, for example, [11,Theorem 84.1]).
If G is a nice-elongation of A by N , we show that if N is countable and G is n-summable, then A is also n-summable (Theorem 1.1); and conversely, if N is Honda (as a valuated group using the height valuation from G), p ω1 G = {0} and A is n-summable, then G is n-summable (Theorem 1.2).It follows that if G is a nice-ℵ 0 -elongation of A and p ω1 G = {0}, then G is n-summable iff A is n-summable (Corollary 1.3).
If A is a separable group and G is an elongation of A by X, then X will always be nice in G, so that an elongation of A is always a niceelongation.Next, if X = p ω G (so that A ∼ = G/p ω G is separable), we will say that G is an ω-elongation of A (this terminology agrees with [20]), so that any ω-elongation is also a nice-elongation; and if X = p ω G is also countable, we will say G is an ω-ℵ 0 -elongation of A.
The group G will be said to be countable plus separable (or cps for short), if it is isomorphic to C ⊕ S, where C is countable and S is separable.Note that if G = D ⊕ R, where D is divisible and R is reduced, then G is cps iff D is countable and R is cps; we may occasionally, therefore, assume that some cps group is actually reduced.We will say that the separable group A has the cps-elongation property if whenever the group G is a (nice-)ℵ 0 -elongation of A, then G is cps; similarly we will say A has the ω-cps-elongation property if whenever the group G is an ω-ℵ 0 -elongation of A, then G is cps.It is straightforward to show that any Σ-cyclic group has the cps-elongation property and that any group with the cps-elongation property has the ω-cps-elongations property (Proposition 3.1); the natural question is whether a group with one of these latter two properties must actually be Σ-cyclic.We show that these questions are independent of the standard set-theoretic axioms (ZFC) by showing that they are consequences of the axiom of constructibility (V = L), but that counter-examples can be constructed using Martin's Axiom and the denial of the Continuum Hypothesis (MA + ¬CH, Theorem 3.4).
We then apply these results to question of when a group G has a nice basis; that is, an ascending sequence of nice subgroups [2]).In particular, we note that MA + ¬CH implies that there is separable group A which is not Σ-cyclic with the property that whenever G is an ω-ℵ 0 -elongation of A by X, then G has a nice basis (Example 4.7).
Referring to [6], a group homomorphism ϕ : G → A is said to be ω 1 -bijective if both the kernel and the co-kernel of ϕ are countable.Thus, if there exists an ω 1 -bijective homomorphism ϕ : G → A, then G/ ker ϕ ∼ = ϕ(G) and G is an ℵ 0 -elongation of ϕ(G) by ker ϕ.In particular, if ker ϕ is a countable nice subgroup of G and ϕ is surjective, then G is a nice-ℵ 0 -elongation of A; so there is a natural connection between these two concepts.We also establish some results pertaining to ω 1 -bijective homomorphisms of special classes of abelian groups (e.g., Proposition 3.6 and Theorem 5.6).

n-Summable Groups
A subgroup X of a valuated group Y is nice if every coset y + X has an element of maximal value; such a y is called proper with respect to X; and in this case, letting v(y + X) = v(y) makes Y /X into a valuated group.
Theorem 1.1.Suppose n is a positive integer and G is a nice-ℵ 0 -elongation of A by N .If G is n-summable, then A is n-summable.
Proof: Fix a valuated decomposition G[p n ] = ⊕ i∈I C i , where each C i is a countable valuated group.We claim there is a countable subset J ⊆ I such that: (a) If we have constructed a countable subset J k of I, we want to construct another countable subset J k+1 of I containing J k such that: (c) for every For a second, fix x to be an element of the countable subgroup N + ⊕ i∈J k C i .We claim that there is countable sequence of elements y This (essentially well-known) fact is a consequence of the assumption that G[p n ] is a valuated direct sum of countable valuated groups and such an object will always be complete in the (induced) ω 1 -topology: To see this, note that if it failed, we could, for all α < ω 1 , choose then z α,i must eventually be constant for each i ∈ I; say it takes on the value w i .We next verify that all but finitely many of the w i must be 0: If this failed, and ω 1 > α > ht(w i ) for an infinite number of indices i, it would easily follow that z α,i = 0 for this same infinite set of indices i, which cannot be.It follows that w = (w i ) i∈I is in ⊕ i∈J C i , and that for all α < ω 1 , ht(x + w) = ht(x + y α + w − y α ) ≥ α, so that x + w = 0; which means that we could let y x,ℓ = w for all ℓ < ω.
We then let J k+1 be a countable subset of I containing J k such that y x,ℓ ∈ ⊕ i∈J k+1 C i for all x ∈ N + ⊕ i∈J k C i and ℓ < ω.
If we then let J = ∪ k<ω J k , then (a) is immediate and (b) follows from (c).
Let L = I − J.If x is a non-zero element of ⊕ i∈L C i , we claim that ht G (x) = ht G/N (x + N ): Note that if this failed, then since N is nice in G, there would be an element y ∈ N such that ht G (x + y) > ht G (x). Now, using (b), we could then find a z ∈ ⊕ i∈J C i such that . By Theorem 2.6 of [8], we will be done if we can show that A[p n ]/V is countable.To that end, suppose P is a countable pure subgroup of G containing N .Note that if z + N ∈ A[p n ], then p n z ∈ N ⊆ P , so p n z = p n w for some w ∈ P .Therefore, z − w ∈ G[p n ] = (⊕ i∈J C i ) ⊕ V , so z − w = t + u where t ∈ ⊕ i∈J C i and u ∈ V .It follows that z = (w+t)+u ∈ (P +⊕ i∈J C i )+V , so that A[p n ] ⊆ (P +⊕ i∈J C i )/N +V , and since (P + ⊕ i∈J C i )/N is clearly countable, A[p n ]/V must be countable, as well.
Recall that a countable valuated group is always Honda, so the following is essentially a converse of the above.
Theorem 1.2.Suppose n is a positive integer, G is a nice-elongation of A by N , p ω1 G = {0} and N is Honda (as a valuated group using the height valuation from G).If A is n-summable, then G is n-summable.
Proof: Since the value spectrum of N is countable, there is a countable ordinal λ such that where the values in X[p n ] are λ plus the heights of elements computed in X (cf.[8, Lemma 1.9]).Since N (λ) = N ∩ p λ G = {0} and N is nice, the map p λ G → p λ A is an isomorphism, so that p λ G, and hence X, is n-summable.We therefore only need to show that G ′ is n-summable.Towards this end, note that G ′ is isotype in G and N is nice in G ′ .It follows that A ′ = G ′ /N embeds as an isotype subgroup of A = G/N .Since p λ+n A ′ = p λ+n (G ′ /N ) = [p λ+n G ′ + N ]/N = {0}, it follows that A ′ [p n ] is Honda.Without loss of generality, then, assume G ′ = G has countable length µ = λ + n and G/N = A is n-Honda.
Let M be the subgroup of G containing N satisfying M/N = (G/N )[p n ] = A[p n ].The height function on G gives a valuation on M and M/N is the valuated direct sum ⊕ i∈I V i , where V i is countable.For each coset in M/N , choose an element which is proper with respect to N and let M p ⊆ M be the collection of all these proper elements.
Let O j for j < ω be an ascending sequence of finite sets of ordinals with union µ.
For each i ∈ I, suppose F i,j , for j < ω is an ascending chain of finite subgroups of V i whose union is all of V i such that the value spectrum Suppose N is the ascending union of the subgroups X j such that the value spectrum of X j is contained in O j .Define We claim that M is the ascending union of these Z j , and that the value spectrum of and since this happens for all i and k, it follows that M/N = [∪ j<ω Z j ]/N , which implies the claim.
where x ∈ X j , and for 1 ≤ ℓ ≤ m, each i ℓ ∈ I and for some , and so we actually assume each g i ℓ is proper with respect to M .Note It follows from Claims 2 and 3 that M is Honda.We now observe that G[p n ] is contained in M , and since M is Honda, so is G[p n ], so that G must be n-summable, as required.
Since a countable valuated group is always Honda, the following is an immediate consequence of Theorems 1.1 and 1.2: Example 1.4.The hypothesis of niceness is necessary in Theorem 1.1.
As in Example 6.8 of [6], let G 0 be a group satisfying the following: there is an embedding of the torsion completion B in G 0 such that Now, suppose M is a countable group such that there is an isomorphism f : , where the first term of this decomposition is divisible and the last two terms are Σ-cyclic and hence separable, (d) then follows.
Example 1.5.Theorem 1.1 does not hold if the nice subgroup N is only assumed to be Honda, as opposed to countable.Suppose A is any separable group which is not Σ-cyclic and N → G is a pure-projective resolution of A (i.e., G is Σ-cyclic and N is pure in G), then N is clearly Honda (since it is Σ-cyclic and the height valuation on G and N agree), G is summable, but A is not summable.
Example 1.6.The hypothesis Suppose G is a totally projective group with N = p ω1 G countably infinite and A = G/N .Then A is a dsc-group, and hence A is summable.Since p ω1 G = {0}, however, G will not be summable.
Example 1.7.The hypothesis of niceness is necessary in Theorem 1.2.
In Example 2.3 of [6], a separable group G was constructed which was not Σ-cyclic but had a (pure) countable subgroup X such that A = G/X was a dsc-group of length ω + 1.It follows that this A is summable, but G is not; in fact G is p ω+1 -projective.

Totally projective groups and generalizations
A nice composition series for the valuated group Y is an ascending chain of nice subgroups It is reasonably easy to verify that if Y has a nice composition series, then so does Y /Y (α) for every ordinal α, and that if Y is Honda (so in particular, if Y is countable), then it has a nice composition series.It is well known that a reduced group G using the height valuation has a nice composition series iff it is totally projective (see, for example, [11,Theorem 81.9]).The following (essentially well-known) observation follows as a direct consequence.Proposition 2.1.Suppose G is a nice-elongation of A by N .If N has a nice composition series as a valuated group and A is totally projective, then G is totally projective.
Proof: If {X i } i≤δ is a nice composition series for N , it can readily be checked that each X i is also nice in G.If {Z j } j≤ǫ is a nice composition series for A = G/N , and for j < µ we let Z ′ j be the subgroup of G containing N defined by the equation Z ′ j /N = Z j , then each Z ′ j will also be nice in G.It readily follows that {X i } i≤δ ∪ {Z ′ j } j≤ǫ is a composition series for G, so G is totally projective.
We include the following (also well-known) observation for future reference.
Corollary 2.2.Suppose G is a reduced group which is a nice-ℵ 0 -elongation of A by N .Then A is totally projective iff G is totally projective.
Proof: Sufficiency follows immediately from Proposition 2.1, and necessity follows directly from Proposition 1.1 of [6] (the second statement does not require the niceness of N in G).
Recall that if λ ≤ ω 1 , then G is a C λ -group if for every α < λ, one (and so each) p α -high subgroup H of G is a dsc-group.If λ is a limit, this is equivalent to requiring that G/p α G is a dsc-group for all α < λ, and if λ is isolated, this is equivalent to requiring that one (and so each) p λ−1 -high subgroup H of G is a dsc-group (see, for example, the discussion in the first two paragraphs of Section 1 in [17]).
Theorem 2.3.Suppose λ ≤ ω 1 , G is a nice-elongation of A by N and N has a nice composition series (using the height function on G).
Proof: Assume A is a C λ -group.First, suppose λ is a limit.Now, for all α < λ there is a nice short-exact sequence Since A/p α A is a dsc-group, it follows from Proposition 2.1 that G/p α G must be a dsc-group.Since this holds for all α < λ, it follows that G is a C λ -group, as required.
Next, suppose λ = γ + 1 is isolated; so, in particular, λ is countable.If λ is finite, the result is trivial (since then any group is a C λ -group), so suppose λ is infinite.We therefore have a nice short-exact sequence: Without loss of generality, then, replace N , G and A by N/N (λ), G/p λ G and A/p λ A, so that we may assume all these groups have length at most λ.
Let Y be a p γ -high subgroup of A, then since γ is infinite, we have that A/Y is divisible and is an isomorphism (see, for example, [14,Theorem 92]).
Let X be the subgroup of G containing N such that can be factored as follows: and since these maps are all surjective, it follows that X is p λ -pure in G (see, for example, [14,Theorem 91]).This means that X is isotype in G, so that Since X ′ will be isotype in X, it follows that X ′ is a dsc-group (see, for example, [14, Theorem 104]).However, this implies that G is a C λ -group, as required.
3 implies that G is as well; and conversely, if A is a C λ -group, then Theorem 3.5 of [6] implies that G is as well (the second implication does not require the niceness of N in G).
Remark 1.If λ > ω, then Example 1.5 shows that in Corollary 2.4, if N is only assumed to be Honda, as opposed to countable, then A may be a C λ -group, while G is not.
For the remainder of this section n will denote a fixed positive integer; G is said to be an n-Σ-group if G[p n ] is the ascending union of a sequence of subgroups [6], a group is an n-Σ-group iff it is a C ω+n -group.Therefore, these last two results can be reformulated as follows: Corollary 2.5.Suppose G is a nice-elongation of A by N .
(a) If N has a nice composition series (using the height function on G) and A group G is said to be pillared if G/p ω G is Σ-cyclic.Since this is true iff for all n < ω, G/p ω+n G is a dsc-group, it follows that a group is pillared iff it is a C ω•2 -group.Letting λ = ω • 2, then, we have the following: Corollary 2.6.Suppose G is a nice-elongation of A by N .
(a) If N has a nice composition series (using the height function on G) and A is pillared, then G is pillared.
A group G is said to be p ω+n -projective if there is a subgroup P ⊆ G[p n ] such that G/P is Σ-cyclic (e.g., [21]).By a classical result of Fuchs (see [13]), if G 1 and G 2 are p ω+n -projective, then G 1 and G 2 are isomorphic iff G 1 [p n ] and G 2 [p n ] are isometric.On the other hand if P is any valuated group with p n P = {0} which is separable in the sense that P (ω) = {0}, then there is a separable group G containing P such that the valuation on P agrees with the height function on G, G is separable and G/P is Σ-cyclic; note that this G will be p ω+n -projective.
More generally, imitating [3], a group G is called p ω+n -totally projective if p ω G is totally projective and G/p ω G is p ω+n -projective.Clearly both p ω+n -projective groups and totally projective groups are themselves p ω+n -totally projective.By the same token, we define a group G to be p ω+n -summable if p ω G is summable and G/p ω G is p ω+n -projective.Observe that p ω+n -projective groups are p ω+n -summable, whereas summable groups need not be p ω+n -summable since there exists a summable group with unbounded torsion-complete first Ulm factor and it is known that torsion-complete p ω+n -projective groups are bounded (see, for instance, [16]).
Proposition 2.7.Suppose G is a nice-ℵ 0 -elongation of A by N .We then have: (a) G is p ω+n -totally projective iff A is p ω+n -totally projective.
Proof: Observe that p ω G is a nice-ℵ 0 -elongation of p ω A by N (ω) = N ∩ p ω G, and G/p ω G is a nice-ℵ 0 -elongation of A/p ω A by N/N (ω).By Theorem 4.2 of [6], G/p ω G is p ω+n -projective iff A/p ω A is p ω+n -projective.Therefore, (a) follows from Corollary 2.2 applied to p ω G and p ω A, and (b) follows from Corollary 1.3 applied to the same.Proof: In (a), if A is Σ-cyclic and G is an ℵ 0 -elongation of A by X, then there is clearly a countable pure subgroup Y of G containing X such that Y /X is a summand of A. Since G/Y ∼ = A/(Y /X) will be Σ-cyclic, it follows that G ∼ = Y ⊕ S, for some Σ-cyclic group S, so that G is cps.It follows that A has the cps-elongation property.The implication in (b) is obvious.

CPS groups and ω 1 -separable groups
We now translate a notion that has been of considerable importance in the study of (torsion-free) free groups (see [10]) into the language of valuated vector spaces.We will say the valuated vector space V is ℵ 1 -coseparable if it is separable (i.e., V (ω) = {0}), and for all subspaces W of V with V /W countable there is a closed subspace U of V such that U ⊆ W and V /U is countable -note that this implies that the quotient valuated vector space V /U is separable and hence free, so that V is isometric to U ⊕ F where F is countable and free.Note also that if V /W is unbounded (i.e., (V /W )(m) = {0} for all m < ω), then the same will be true of F .
We pause to review the following standard construction (see, for example, [14, Theorem 106]).
Proof: Let Y be a pure and dense subgroup of A such that Y [p] = D, so that A/Y is divisible; and let Z be a divisible hull for X, so that Z/N is also divisible.The isomorphism A[p]/D ∼ = X/pX implies that there is an isomorphism φ : A/Y → Z/X, and we can let G = {(a, z) ∈ A ⊕ Z : f (a+Y ) = z +X}.The assignment x → (0, x) gives an injection X → G, and the assignment (a, z) → a gives a surjection G → A, and it can be checked that these conditions will imply the conclusions.
Proof: It is clear that (a) implies (b).Suppose next that (b) holds and that V is ℵ 1 -coseparable; we want to show that V must be free.Let A be a separable group containing V as a subgroup where A/V is Σ-cyclic, so that A is p ω+1 -projective.Since there is a valuated injection of the valuated quotient space A[p]/V into (A/V )[p] and the latter is Honda, so is A[p]/V , and hence free as a valuated vector space.It follows that A[p] is isometric to V ⊕ F where F is a free valuated vector space.In fact, after possibly adding to A a summand which is an infinite Σ-cyclic group, we may assume that every Ulm invariant of F is infinite.We now show that this A has the ω-cps-elongation property.To this end, let G be an ω-ℵ 0 -elongation of A, so that A = G/p ω G; we need to show G is cps.We begin by letting preserves all finite heights, it follows that P also maps isometrically onto U .
From this construction we can conclude that [ Since p ω G∩P = {0}, the first statement readily follows.Since G/Q ∼ = A/U , we must show that the latter is Σ-cyclic.Observe that V /U is a free, separable valuated vector space, so there is a valuated embedding V /U → J, where J is a Σ-cyclic group using the height valuation.Since V is nice in A and A/V is Σ-cyclic, it follows that this embedding extends to a homomorphism f : A → J, for which the kernel of f V is U .It follows that the kernel of the obvious map A → (A/V ) ⊕ J is U , and since (A/V ) ⊕ J is Σ-cyclic, so is A/U .
Observe that if m < ω, then f G,Q (m) = f A,U (m).Since F has infinite Ulm invariants, U ⊆ V , and A[p] is isometric to V ⊕ F , the claim follows.
Let C be a countable group whose Ulm function is defined by Note this implies that p ω C and p ω G are isomorphic, even when they are not reduced.Let H = C ⊕ A, so that In addition, Q ′ and Q are both isometric to p ω G ⊕ U and so they are isometric to each other.Next, it is readily checked that Therefore, by the fundamental result of Hill on extending isometries on nice valuated subgroups of groups (see, for example, [11,Theorem 83.4]), it follows that there is an isomorphism G → H = C ⊕ A. It therefore follows that G is cps, so that A has the ω-cps-elongation property.By hypothesis, then, A must be Σ-cyclic, and this implies that A[p] is free as a valuated vector space, and since V is a subspace of A[p], it is also free, as required.
We now need to show that (c) implies (a), so suppose every ℵ 1 -coseparable valuated vector space is free and A is a separable group with the ω-cps-elongation property, and let Note that W is closed (and hence, nice) in W ′ and W ′ /W is countable (and hence, free), so that W ′ = W ⊕ F for some countable, free subspace F .
Note that By hypothesis, G = C ⊕ S, where C is countable and S is separable.First, note that we can identify , so that we can view S[p] as a subgroup of W ′ ⊆ A[p] = V .We now let U = S[p] ∩ W . Since both V /S[p] and V /W are countable, and V /U embeds in (V /S[p]) ⊕ (V /W ), it follows that V /U is countable.
To show that U is closed, let u i for i < ω be a sequence in U which converges in the p-adic topology to z ∈ V .Since Note that the arguments in the last two paragraphs imply that V is ℵ 1 -coseparable.It follows, therefore, that V = A[p] is free, so that A is Σ-cyclic, as required.
Theorem 3.4.The following hold: (a) Assuming (MA + ¬CH), there is a separable group A which has the cps-elongation property (and hence it also has the ω-cps-elongation property), but is not Σ-cyclic.(b) Assuming (V = L), if A is a separable group which has the ω-cpselongation property (in particular, if A has the cps-elongation property), then A is Σ-cyclic.
Before proceeding, note that the last result implies the following.Recall that a group A is ω 1 -separable iff it is separable and every countable subgroup of A is contained in a countable summand of A. Using a (by now) standard construction, it can be shown in ZFC that there are ω 1 -separable groups of cardinality ℵ 1 which are not Σ-cyclic.
Proof of Theorem 3.4: We claim that C will be pure in G: Note that C/X is clearly pure in G/X, which implies that C/P ∼ = (C/X)/(P/X) is pure in G/P ∼ = (G/X)/(P/X).Since P is pure in G, it follows that C is pure in G, as required.
If we let A = A ′ ⊕ (C/X), then the countability of C/X and the fact that A is ω 1 -separable readily implies that A ′ is also ω 1 -separable.In the presence of MA + ¬CH, by Theorem 3.1 of [18], we can conclude that Pext(A ′ , C) = {0}.However, since C is pure in G and G/C ∼ = (G/X)/(C/X) ∼ = A ′ , we can conclude that G ∼ = A ′ ⊕ C, as required.
Regarding (b), note that if G is an ω-elongation of A by Z p , then since A has the ω-cps-elongation property, it follows that G ∼ = C ⊕ S, where C is countable and S is separable.If H ω+1 is the generalized Prüfer group of length ω + 1, there is clearly a homomorphism G → C → H ω+1 which is non-zero on p ω G = p ω C. In the presence of V = L, by Theorem 2.2 of [20], A must be Σ-cyclic.
Although by Proposition 7.2 of [4], A being ω 1 -separable implies that A/C is ω 1 -separable whenever C is a countable nice subgroup of A, we shall see that the converse implication does not hold in some versions of set theory.First, we pause for the following assertion which answers Problem 7.3 from [4].Proposition 3.6.Suppose A is a separable group and f : Proof: Let I be the image of f and K be the kernel of f .Since I is a subgroup of A, K is a countable and nice subgroup of G, so by Proposition 7.2 of [4], I ∼ = G/K is ω 1 -separable.Now, if C is a countable subgroup of A, then after possibly expanding C, we may assume that A = I + C. Let C ′ be a countable subgroup of I containing I ∩ C such that I = I ′ ⊕ C ′ ; the existence of such a C ′ follows from the fact that I is ω 1 -separable.Then C ⊆ C + C ′ is countable and one easily checks that A = I ′ ⊕ (C ′ + C).It follows that A is ω 1 -separable, as desired.
The next result shows that the converse of Proposition 3.6 is undecidable in ZFC for groups of cardinality ℵ 1 .It settles Problem 7.2 from [4].
Proof: Regarding (a), a separable group H is weakly ω 1 -separable if for every countable subgroup C of H, the p-adic closure C is also countable.By [19], in the presence of MA + ¬CH, for groups of cardinality ℵ 1 , the classes of weakly ω 1 -separable and ω 1 -separable groups coincide.Finally, in view of Corollary 5.2 of [6] we have that G is weakly ω 1 -separable iff A is weakly ω 1 -separable.Regarding (b), Megibben [19,Theorem 3.2] found in the presence of V = L an ω 1 -separable group A of cardinality ℵ 1 containing a pure and dense subgroup G ′ which is not ω 1 -separable such that A/G ′ is countable.If C is a countable subgroup of A such that G ′ + C = A, then let G = G ′ ⊕C (note this is an external direct sum).Since G ′ is not ω 1 -separable, the countability of C easily implies that G is not ω 1 -separable.If we then consider the sum map G = G ′ ⊕ C → G ′ + C = A, defined by (g, c) → g + c, we have our result.

Groups with nice bases
Following [2], recall that a nice basis for a group G is an ascending sequence of nice subgroups and all G i /N are nice in G/N and are Σ-cyclic groups.Therefore, G = ∪ i<ω G i with p ω G i ⊆ N .Hence p ω G i ⊆ N ∩p ω G = {0} and we conclude that all G i are separable.By Corollary 2.2, they are Σ-cyclic groups.Moreover, utilizing Lemma 79.3 of [11], we derive that these G i are also nice in G, as needed.
The last result can also be deduced from [4, Proposition 9.1]; nevertheless we have given another, more smooth, proof.Suppose A is any separable thick group and let G be any group such that p ω G is countable, reduced inseparable (thus it is not Σ-cyclic) with G/p ω G ∼ = A. Applying Theorem 1.3 of [7], G does not has a nice basis (see also Proposition 4.5 below).Let G be a reduced group without a nice basis with a countable basic subgroup N (for example, if G/p ω G is torsion-complete with a countable basic subgroup and p ω G is not Σ-cyclic, it follows from [7] that G does not have a nice basis).Clearly, N (ω) = {0} is satisfied and A = G/N is divisible, hence it does have a nice basis owing to [2].
We now investigate the hypothesis that N (ω) = {0} in the last result by contrasting it with the case where G is an ω-ℵ 0 -elongation of A by N ; i.e., we are contrasting the situation in which N ∩ p ω G = {0} with the situation in which N = p ω G.We begin by considering an apparently unrelated notion.
A separable valuated vector space V (i.e., V (ω) = {0}) will be called essentially finitely indecomposable (or efi for short) if (1) it is unbounded (in the sense that V (m) = {0} for all m < ω); and (2) there is no valuated decomposition V = W ⊕F where F is an unbounded free valuated vector space.For example, if V is complete in the p-adic topology (i.e., if it is isometric to the socle of a torsion-complete group), then V is efi.The following topological characterization utilizes ideas from [7].Proposition 4.4.An unbounded separable valuated vector space V is efi iff V cannot be expressed as the ascending union of a sequence of closed subspaces Conversely, if we have the collection of closed subgroup C m satisfying the above, then let x m ∈ V − C m be chosen so that v(x m ) ≥ m and x m is a valuated summand of V /C m ; denote the valuated projection then it is easy to verify that f is a valuated homomorphism whose image is unbounded and is actually contained in the free valuated vector space ⊕ m<ω x m .If we let W be the kernel of f , then V = W ⊕F , where F is an unbounded free valuated vector space.Proposition 4.5.If A is any unbounded separable group that has a subsocle P ⊆ A[p] which is efi, then there is a group G which is an ω-ℵ 0 -elongation of A, such that G does not have a nice basis.
Proof: Let N be any reduced countable group which is not Σ-cyclic, i.
then L i is closed in the p-adic topology on A and their union is all of A. It follows that P is the union of the closed subspaces C i = P ∩ L i .Since P if efi, it follows from Proposition 4.4 that, for some i, P (i) ⊆ P ∩ C i ⊆ L i .We therefore have A[p] ⊆ D + L i .
If x ∈ N , then find y ∈ G such that py = x.Since y + N ∈ A[p] there is a w ∈ G[p] and z ∈ M i such that y + N = (w + z) + N .So there is a u ∈ N such that y = w + z + u.This implies that x = py = pw + pz + pu = pz + pu.Note that pz ∈ M i (ω), pu ∈ pN , so that we can conclude N = M i (ω) + pN .This, in turn, implies that N/M i (ω) is divisible.Since M i is nice in G, M i (ω) is nice in N .Since N is reduced and N/M i (ω) is divisible, we must have that N = M i (ω) ⊆ M i .This is a contradiction, because M i being a part of a nice basis is Σ-cyclic, though we assumed N was not.
If P is any separable valuated vector space which is efi, then there is a separable group A containing P such that the valuation on P agrees with the height function on A and A/P is Σ-cyclic.
Remark 2. If we start with the group A from Example 4.6 and construct the group G as in Proposition 4.5, then since G does not have a nice basis, it follows from [2] that it is not p ω+1 -projective.Therefore, in Theorem 4.2 of [6], the condition of separability is necessary and cannot be eliminated.Following [7], the separable group A is said to have the nice basis extension property if for all ω-elongations G of A, if p ω G has a nice basis, then so does G.By Corollary 2.13 of [7], the Continuum Hypothesis (CH) implies that when A has a countable basic subgroup, then A has the nice basis extension property iff it is Σ-cyclic, and it was asked whether all groups with this property are necessarily Σ-cyclic.The following is related to this question by restricting to the case of ω-ℵ 0 -elongations.
Example 4.7.In the presence of MA + ¬CH, there is a separable group A which is not Σ-cyclic, with the property that every ω-ℵ 0 -elongation G of A has a nice basis.This group may be chosen to be p ω+1 -projective.
Let A be as in Theorem 3.4(a).If G is an ω-elongation of A, then since A has the ω-cps-elongation property, G ∼ = C ⊕ S, where C is countable and S is separable.Since C is countable, it is totally projective, and hence it has a nice basis by [2].Since S is separable, it clearly has a nice basis, as well.Therefore, in view of [2], G has a nice basis, as required.The last part follows from Theorem 3.3.
The following completely answers Problem 4 of [7].
Example 4.8.There are groups G 1 and G 2 with p ω G 1 ∼ = p ω G 2 and G 1 /p ω G 1 ∼ = G 2 /p ω G 2 such that G 1 has a nice basis but G 2 does not have a nice basis.
Suppose A ′ is a separable group with a subsocle P which is efi, and B is an unbounded Σ-cyclic group.So if A = A ′ ⊕ B, then since P is also a subsocle of A, by Proposition 4.5, there is an ω-ℵ 0 -elongation G 2 of A by a countable group N = p ω G 2 which does not have a nice basis.Since N is countable and B is unbounded, there a dsc-group H such that p ω H ∼ = N and H/p ω H ∼ = B.If we let G 1 = H ⊕ A ′ , then H has a nice basis by [2] (since it is a dsc-group and hence totally projective), A ′ has a nice basis (since it is separable), so that G 1 has a nice basis in virtue of [2], as required.
Question 1.Is the converse of Proposition 3.1(b) valid in ZFC?That is, if A is a separable group with the ω-cps-elongation property, does it follow in ZFC that A has the cps-elongation property?Question 2. Suppose A is a separable group with the property that every ω-ℵ 0 -elongation G of A has a nice basis.In V = L, does it follow that A must be Σ-cyclic?(Note that by Example 4.7, this does not hold in a model of MA + ¬CH.)Question 3. Suppose A is a separable group with the property that every ω-elongation G by a totally projective group X = p ω G has a nice basis.Can we conclude that A is Σ-cyclic?(If this holds, then all such G will also be totally projective.)

Almost totally projective groups
A group G is almost totally projective if it has a collection of nice subgroups N with the properties: (0) {0} ∈ N .(1) If {H i } i∈I is an ascending chain of subgroups in N , then ∪ i∈I H i ∈ N .
(2) If C ⊆ G is countable, then there is a countable M ∈ N such that C ⊆ M .
We pause to review the following.

Proposition 3 . 1 .
Suppose A is a separable group.(a) If A is Σ-cyclic, then it has the cps-elongation property.(b) If A has the cps-elongation property then it has the ω-cps-elongation property.

Corollary 3 . 5 .
The following statements are independent of ZFC: (a) Every separable group with the cps-elongation property is Σ-cyclic.(b) Every separable group with the ω-cps-elongation property is Σcyclic.(c) Every ℵ 1 -coseparable valuated vector space is free.Proof: By Theorem 3.4, both (a) and (b) fail in a model of MA + ¬CH and both are true in V = L.By Theorem 3.3, (b) is equivalent to (c).

Proposition 4 . 1 .
Let G be a nice-ℵ 0 -elongation of A by N such that N ∩ p ω G = {0}.If A has a nice basis, then G has a nice basis.

Example 4 . 3 .
In Proposition 4.1, the niceness of N in G is also necessary.
e., p ω N = {0}.Let D be a dense subspace of A[p] such that A[p]/D is countably infinite and D + P (i) = A[p] for all i < ω [start with a dense subspace D ′ of P such that P/D ′ is countably infinite, and if W is a subspace of A[p] such that P ⊕ W = A[p], then we can let D = D ′ + W ]. By Lemma 3.2 there is a group G which is an ω-elongation of A by N such that D = [G[p] + p ω G]/p ω G ⊆ A[p].Suppose now that G actually does have a nice basis M [5]position 5.1 ([15],[5]).If K is a subgroup of the reduced group H such that H/K is countable and K is almost totally projective, then H is almost totally projective.Proposition 5.2 ([5]).If K is a countable and nice subgroup of the reduced group H such that H/K is almost totally projective, then H is almost totally projective.