On properties of n-totally projective abelian p -groups
We prove some properties of $n$-totally projective abelian $p$-groups. Under some additional conditions for the group structure, we obtain an equivalence between the notions of $n$-total projectivity and strong $n$-total projectivity. We also show that $n$-totally projective $A$-groups are isomorp...
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| author | Danchev, P. V. Keef, P. Данчев, П. В. Кеф, П. |
| author_facet | Danchev, P. V. Keef, P. Данчев, П. В. Кеф, П. |
| author_sort | Danchev, P. V. |
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| description | We prove some properties of $n$-totally projective abelian $p$-groups.
Under some additional conditions for the group structure, we obtain an equivalence between the notions of $n$-total projectivity and strong $n$-total projectivity.
We also show that $n$-totally projective $A$-groups are isomorphic if they have isometric $p^n$-socles. |
| first_indexed | 2026-03-24T02:26:53Z |
| format | Article |
| fulltext |
UDC 512.5
P. Keef (Whitman College, USA),
P. Danchev (Plovdiv Univ. „P. Hilendarski”, Bulgaria)
ON PROPERTIES OF n-TOTALLY PROJECTIVE ABELIAN p-GROUPS
ПРО ВЛАСТИВОСТI n-ТОТАЛЬНО ПРОЕКЦIЙНИХ АБЕЛЕВИХ p-ГРУП
We prove some properties of n-totally projective abelian p-groups. Under some additional conditions for the group structure,
we obtain an equivalence between the notions of n-total projectivity and strong n-total projectivity. We also show that
n-totally projective A-groups are isomorphic if they have isometric pn-socles.
Доведено деякi властивостi n-тотально проекцiйних абелевих p-груп. При деяких додаткових умовах на будову
груп встановлено еквiвалентнiсть понять n-тотальної проективностi та сильної n-тотальної проективностi. Також
показано, що n-тотально проективнi A-групи iзоморфнi, якщо вони мають iзометричнi pn-цоколi.
Introduction. Throughout this paper, let us assume that all groups are additive p-primary groups
and n is a fixed natural. Foremost, we recall some crucial notions from [7] and [8] respectively.
Definition 1. A group G is said to be n-simply presented if there exists a pn-bounded subgroup
P of G such that G/P is simply presented. A summand of an n-simply presented group is called
n-balanced projective.
Definition 2. A group G is said to be strongly n-simply presented = nicely n-simply presented
if there exists a nice pn-bounded subgroup N of G such that G/N is simply presented. A summand
of a strongly n-simply presented group is called strongly n-balanced projective.
Clearly, strongly n-simply presented groups are n-simply presented, while the converse fails (see,
e.g., [7]).
Definition 3. A group G is called n-totally projective if, for all ordinals λ, G/pλG is pλ+n-
projective.
Definition 4. A group G is called strongly n-totally projective if, for any ordinal λ, G/pλ+nG
is pλ+n-projective.
Apparently, strongly n-totally projective groups are n-totally projective, whereas the converse
is wrong (see, for instance, [8]). Moreover, (strongly) n-simply presented groups are themselves
(strongly) n-totally projective, but the converse is untrue (see, for example, [8]).
Definition 5. A group G is called weakly n-totally projective if, for each ordinal λ, G/pλG is
pλ+2n-projective.
Evidently, n-totally projective groups are weakly n-totally projective with the exception of the
reverse implication which is not valid.
The purpose of the present article is to explore some critical properties of n-totally projective
groups, especially when some of the three variants of n-total projectivity do coincide. In fact, we
show that if the group G is an A-group, then the concepts of being n-totally projective and strongly
n-totally projective will be the same (Theorem 1). However, this is not the case for weakly n-
totally projective groups (Example 1). We also establish that two n-totally projective A-groups are
isomorphic if and only if they have isometric pn-socles, i.e., isomorphic socles whose isomorphism
preserves heights as computed in the whole group (Corollary 1). Likewise, we exhibit a concrete
example of a strongly n-totally projective group with finite first Ulm subgroup that is not ω + n-
c© P. KEEF, P. DANCHEV, 2012
766 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON PROPERTIES OF n-TOTALLY PROJECTIVE ABELIAN p-GROUPS 767
totally pω+n-projective (Example 2). Finally, some assertions about (strongly) n-simply presented
and n-balanced projective groups are obtained as well (Proposition 3 and Corollaries 2 – 4).
We note for readers’ convenience that all undefined explicitly notations and the terminology are
standard and follow essentially those from [2 – 4]. Besides, for shortness, we will denote the torsion
product Tor (G,H) of the groups G and H by G5H . Also, for any group G and ordinal λ, LλG is
its completion in the pλ-topology and let EλG = (LλG)/G.
Main results. We begin here with the equivalence of strong n-total projectivity and n-total
projectivity under the extra assumption that the full group is an A-group. Specifically, the following
holds:
Theorem 1. Suppose G is an A-group. Then the following three conditions are equivalent:
(a) G is n-totally projective;
(b) G is strongly n-totally projective;
(c) for every limit ordinal λ of uncountable cofinality, we have pnEλG = {0}.
Proof. We first turn to a few thoughts on A-groups introduced in [4]. Let λ be a limit ordinal,
and let
0→ G→ H → K → 0 (1)
be a pλ-pure exact sequence with H a totally projective group of length λ and K a totally projective
group. If λ has countable cofinality or pλK = {0}, then G is also totally projective. Otherwise, G is
said to be a λ-elementary A-group. Note that pλK is naturally isomorphic to (LλG)/G = Eλ where
LλG is the completion in the pλ-topology. An A-group G is then defined to be the direct sum of a
collection of λ-elementary A-groups, for various ordinals of uncountable cofinality. Note that these
groups G are classified in [4] up to an isomorphism using their Ulm invariants, together with the Ulm
invariants of the totally projective groups EλG, over all limit ordinals λ of uncountable cofinality.
Next, since a direct sum of groups is (strongly) n-totally projective if and only if each of its terms
has that property, and since the functor EλG also respects direct sums (because λ has uncountable
cofinality), we may assume that G is a λ-elementary A-group and that we possess a representing
sequence as in (1). Notice that for any limit ordinal β < λ, we have a balanced-exact sequence
implied via (1)
0→ G/pβG→ H/pβH → K/pβK → 0.
On the other hand, since K is totally projective, K/pβK is pβ-projective, so that this sequence splits.
It now follows that G/pβG is a summand of the totally projective group H/pβH , and hence it is
pβ-projective too. Our result will therefore follow from the statement:
Claim. If λ is a limit ordinal of uncountable cofinality and G is a λ-elementary A-group, then
G ∼= G/pλG ∼= G/pλ+nG is pλ+n-projective if and only if pnEλG ∼= pλ+nK = {0}.
In order to prove that Claim, observe that (1) can actually be viewed as a pλ-pure projective
resolution of K. Compare this with the standard pλ-pure projective resolution of K given by
0→Mλ 5K → Hλ 5K → K → 0
where Mλ is a λ-elementary S-group of length λ and Hλ is the Prüfer group of length λ (see [8]).
By virtue of the Schanuel’s lemma (cf. [3]), there is an isomorphism
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
768 P. KEEF, P. DANCHEV
(Mλ 5K)⊕H ∼= (Hλ 5K)⊕G.
Since H and Hλ5K are obviously pλ-projective, it suffices to show that Mλ5K is pλ+n-projective
if and only if pλ+nK = {0}.
To this aim, suppose first that pλ+nK = {0}; so in particular, K is pλ+n-projective, whence
Mλ5K is pλ+n-projective (see [9]). For the converse, we see that Hλ5K will also be complete in
the pλ-topology. Consequently, Eλ(Mλ5K) ∼= EλG ∼= pλK. Supposing pλ+nK 6= {0}, we need to
demonstrate that Mλ 5K is not pλ+n-projective. Considering a direct summand of K, it suffices to
assume that pλK is cyclic of order pm, where m > n. Let M be a pλ-high subgroup of K. It follows
that M is also pλ+n-high in K and hence it is pλ+n+1-pure in K. In addition, K/M ∼= Z(p∞), so
that Mλ 5 (K/M) ∼=Mλ. It would then follow that the sequence
0→Mλ 5M →Mλ 5K →Mλ → 0
is pλ+n+1-pure. If Mλ 5 K actually were pλ+n-projective, then Lemma 2.1 (g) from [8] would
imply that the sequence splits. Therefore, Mλ is isomorphic to a summand of Mλ 5 K. However,
Eλ(Mλ 5K) ∼= pλK is reduced, whereas EλMλ
∼= Z(p∞) is divisible. This contradiction proves
the entire Claim and hence the theorem.
As a consequence, we yield the following result concerning the isomorphism characterization of
n-totally projective A-groups.
Corollary 1. Suppose G and G′ are n-totally projective A-groups. Then G and G′ are isomorphic
if and only if G[pn] and G′[pn] are isometric.
Proof. Applying Theorem 1, G and G′ are both strongly n-totally projective and both EλG,EλG′
are pn-bounded for each limit ordinal λ of uncountable cofinality. Since G and G′ clearly possess
identical Ulm invariants, we need to illustrate that for for any λ as above we have EλG ∼= EλG
′.
It is readily checked that every element of EλG can be represented by a neat Cauchy net {xi}i<α
where each xi ∈ G[pn]. This means that EλG can also be described as Lλ(G[pn])/(G[pn]), where
the numerator of this expression consists of the inverse limit of G[pn]/(pαG)[pn] over all α < λ.
Since G[pn] and G′[pn] are isometric, by what we have shown above it follows that EλG and EλG′
are isomorphic for all λ. But employing [5], we can conclude that G ∼= G′, as claimed.
Corollary 1 is proved.
The following example shows that Theorem 1 is not longer true for weakly n-totally projective
groups.
Example 1. There exists a weakly n-totally projective A-group which is not n-totally projective.
Proof. Construct any A-group G of length ω1 which is proper pω1+2-projective, that is, pω1+2-
projective but not pω1+1-projective. For example, if Mω1 is an elementary S-group of length ω1,
and Hω1+2 is the Prüfer group of length ω1 + 2, then G = Hω1+2 5Mω1 will be such a group.
Furthermore, it follows immediately that G is weakly 1-totally projective but it is not 1-totally
projective as desired.
The next example shows that the class of strongly n-totally projective groups is not contained in the
class of ω+n-totally pω+n-projective groups. Recall that in [1] a group G is said to be ω+n-totally
pω+n-projective group if each pω+n-bounded subgroup is pω+n-projective.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON PROPERTIES OF n-TOTALLY PROJECTIVE ABELIAN p-GROUPS 769
Example 2. There exists a strongly n-totally projective group with finite inseparable first Ulm
subgroup which is not ω + n-totally pω+n-projective.
Proof. Suppose A is a separable pω+1-projective group whose socle A[p] is not ℵ0-coseparable
(such a group exists even in ZFC and is common to construct) and H is a countable group with pωH
being finite and pω+nH 6= 0. Letting G = A ⊕ H , then G is strongly n-totally projective. Indeed,
it is pretty easy to see that G/pλ+nG is pλ+n-projective for any (limit) ordinal λ because both A
and H are n-totally projective. Since G is neither a direct sum of countable groups nor a pω+n-
projective group, if it were ω+ n-totally pω+n-projective, it would be proper. However, appealing to
Theorem 3.1 of [1], this cannot be happen.
Another example in this way can be found in ([6], Example 2.5).
On the other hand, ω + n-totally pω+n-projective groups are contained in the class of n-totally
projective groups. In fact, by a plain combination of Proposition 3.1 and Theorem 1.2 (a1) in [6]
along with [7], ω + n-totally pω+n-projective groups are themselves n-simply presented and thus
they are n-totally projective, as asserted.
In this way the following statement is true as well. Imitating [1], recall that a group is said to be
ω-totally pω+n-projective if every its separable subgroup is pω+n-projective.
Proposition 1. Each n-totally projective group with countable first Ulm subgroup is ω-totally
pω+n-projective.
Proof. If G is n-totally projective, then with the aid of Definition 3 we obtain that the quotient
G/pωG will actually be pω+n-projective, and so ω-totally pω+n-projective. Since pωG is countable
and the ω-totally pω+n-projective groups are closed under ω1-bijections (see [6]), G will be ω-totally
pω+n-projective, as expected.
We will be next concentrated to some characteristic properties of (strongly) n-totally projective
groups.
Proposition 2. Let P ≤ G[p].
(a) If G is (strongly) n-totally projective, then G/P is (strongly) n+ 1-totally projective.
(b) If G/P is (strongly) n-totally projective, then G is (strongly) n+ 1-totally projective.
Proof. We shall prove the statement only for n-totally projective groups since the situation with
strongly n-totally projective groups is quite similar.
(a) If λ is an ordinal and Gλ = G/pλG, then there is an exact sequence
0→ (P + pλG)/pλG→ Gλ → G/(P + pλG)→ 0.
Since p((P + pλG)/pλG) = {0} and Gλ is pλ+n-projective, it follows that H = G/(P + pλG) is
pλ+n+1-projective. However, if Q = (P + pλG)/P ⊆ A = G/P , then Q ⊆ pλA. In addition,
H ∼= (G/P )/((P + pλG)/P ) = A/Q
is pλ+n+1-projective. Moreover, it follows also that
Hλ = H/pλH ∼= A/Q/pλ(A/Q) = A/Q/pλA/Q ∼= A/pλA = Aλ
is pλ+n+1-projective. Note that this implies that A is n+ 1-totally projective, as required.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
770 P. KEEF, P. DANCHEV
(b) Suppose now that A = G/P is n-totally projective. If P ′ = G[p]/P ⊆ A[p], then by what we
have already shown above pG ∼= G/G[p] ∼= (G/P )/(G[p]/P ) = A/P ′ is n + 1-totally projective.
However, this easily forces by [8] that G itself is n+ 1-totally projective, as claimed.
We will now establish some affirmations for n-simply presented groups and their direct summands
called n-balanced projective groups. So, the next few results show that an n-balanced projective
group must be pretty close to being n-simply presented, since they illustrate that the complementary
summand can be chosen in special ways. Recall that a group B will be said to be a BT-group if it is
isomorphic to a balanced subgroup of a totally projective group. It plainly follows that a BT-group is
also an IT-group (i.e., one that is isomorphic to an isotype subgroup of a totally projective group).
Proposition 3. Suppose G is a group of length λ. Then the following hold:
(a) If G is n-balanced projective, then there is a BT-group X with pλX = {0} such that G⊕X is
n-simply presented.
(b) If G is strongly n-balanced projective, then there is an IT-group K with pλK = {0} such that
G⊕K is strongly n-simply presented.
Proof. (a) Using the notation of Theorem 1.2 from [7], we start with a balanced projective
resolution
0→ X → Y → G→ 0,
so that X is a BT -group. Knowing this, we can construct an n-balanced projective resolution
0→ X → Z → G→ 0
of G. Since G is n-balanced projective, we can conclude that G⊕X → Z is n-simply presented, as
required.
(b) Using the notations of Lemma 1.4 and Theorem 1.5 of [7], there is a strongly n-balanced
projective resolution of G given by
0→ K(G)→ H(G)→ G→ 0
where H(G) = K(G[pn]) is strongly n-simply presented. Note that H(G)[pn] is isometric to the val-
uated direct sum G[pn]⊕K(G)[pn]. It follows that K(G)[pn] embeds isometrically in H(G)/G[pn].
Therefore K(G) embeds as an isotype subgroup of H(G)/G[pn], which is obviously totally projec-
tive.
As immediate consequences, we derive the following corollaries.
Corollary 2. Let G be a (strongly) n-balanced projective group of countable length. Then there
exists a direct sum of countable groups X of countable length such that G⊕X is (strongly) n-simply
presented.
Proof. Since IT -groups of countable length are direct sums of countable groups, we may directly
apply Proposition 3.
Corollary 3. Let G be an n-balanced projective group. If the balanced projective dimension of
G is at most 1, then there is a totally projective group X such that G⊕X is n-simply presented.
Proof. Again, if
0→ X → Y → G→ 0
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON PROPERTIES OF n-TOTALLY PROJECTIVE ABELIAN p-GROUPS 771
is a balanced projective resolution of G, then X will be totally projective, and G⊕X will be n-simply
presented.
Corollary 4. Let G and G′ be strongly n-balanced projective groups. If G[pn] is isometric to
G′[pn], so that they have the same length λ, then there are IT -groups K and K ′ of length at most λ
such that G⊕K is isomorphic to G′ ⊕K ′.
Proof. An isometry G[pn] → G′[pn] leads to an isomorphism H(G) → H(G′), and thus the
result follows from Proposition 3 (b).
We close the work with the following three problems:
Problem 1. Find an ω-totally pω+n-projective group which is not n-totally projective, and an n-
totally projective group with a uncountable first Ulm subgroup that is not ω-totally pω+n-projective.
Problem 2. Does it follow that n-simply presented A-groups are strongly n-simply presented?
Problem 3. Does there exist a pω1+1-projective N -group of length ω1 which is not totally projec-
tive, i.e., is not a direct sum of countable groups?
1. Danchev P., Keef P. An application of set theory to ω+n-totally pω+n-projective primary abelian groups // Mediterr.
J. Math. – 2011. – 8, № 4. – P. 525 – 542.
2. Fuchs L. Infinite abelian groups. – New York; London: Acad. Press, 1970, 1973. – Vol. 1, 2.
3. Griffith Ph. Infinite abelian group theory. – Chicago; London: Univ. Chicago Press, 1970.
4. Hill P. On the structure of abelian p-groups // Trans. Amer. Math. Soc. – 1985. – 288, № 2. – P. 505 – 525.
5. Hill P., Megibben C. On direct sums of countable groups and generalizations // Stud. Abelian Groups. – 1968. –
P. 183 – 206.
6. Keef P. On ω1-pω+n-projective primary abelian groups // J. Algebra Numb. Th. Acad. – 2010. – 1, № 1. – P. 41 – 75.
7. Keef P., Danchev P. On n-simply presented primary abelian groups // Houston J. Math. – 2012. – 38, № 3.
8. Keef P., Danchev P. On m,n-balanced projective and m,n-totally projective primary abelian groups (to appear).
9. Nunke R. On the structure of tor II // Pacif. J. Math. – 1967. – 22. – P. 453 – 464.
Received 07.10.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
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| spelling | umjimathkievua-article-26152020-03-18T19:30:57Z On properties of n-totally projective abelian p -groups Про властивостi n-тотально проекцiйних абелевих p -груп Danchev, P. V. Keef, P. Данчев, П. В. Кеф, П. We prove some properties of $n$-totally projective abelian $p$-groups. Under some additional conditions for the group structure, we obtain an equivalence between the notions of $n$-total projectivity and strong $n$-total projectivity. We also show that $n$-totally projective $A$-groups are isomorphic if they have isometric $p^n$-socles. Доведено деякi властивостi $n$-тотально проекцiйних абелевих $p$-груп. При деяких додаткових умовах на будову груп встановлено еквiвалентнiсть понять $n$-тотальної проективностi та сильної $n$-тотальної проективностi. Також показано, що $n$-тотально проективнi $A$-групи iзоморфнi, якщо вони мають iзометричнi$p^n$-цоколi. Institute of Mathematics, NAS of Ukraine 2012-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2615 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 6 (2012); 766-771 Український математичний журнал; Том 64 № 6 (2012); 766-771 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2615/1989 https://umj.imath.kiev.ua/index.php/umj/article/view/2615/1990 Copyright (c) 2012 Danchev P. V.; Keef P. |
| spellingShingle | Danchev, P. V. Keef, P. Данчев, П. В. Кеф, П. On properties of n-totally projective abelian p -groups |
| title | On properties of n-totally projective abelian p -groups |
| title_alt | Про властивостi n-тотально проекцiйних абелевих p -груп |
| title_full | On properties of n-totally projective abelian p -groups |
| title_fullStr | On properties of n-totally projective abelian p -groups |
| title_full_unstemmed | On properties of n-totally projective abelian p -groups |
| title_short | On properties of n-totally projective abelian p -groups |
| title_sort | on properties of n-totally projective abelian p -groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2615 |
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