On uniformly fully inert subgroups of abelian groups

Abstract If H is a subgroup of an abelian group G and φ ∈ End(G), H is called φ-inert (and φ is H-inertial) if φ(H) ∩ H has finite index in the image φ(H). The notion of φ-inert subgroup arose and was investigated in a relevant way in the study of the so called intrinsic entropy of an endomorphism φ, while inertial endo-morphisms (these are endomorphisms that are H-inertial for every subgroup H) were intensively studied by Rinauro and the first named author. A subgroup H of an abelian group G is said to be fully inert if it is φ-inert for every φ ∈ End(G). This property, inspired by the “dual” notion of inertial endomorphism, has been deeply investigated for many different types of groups G. It has been proved that in some cases all fully inert subgroups of an abelian group G are commensurable with a fully invariant subgroup of G (e.g., when G is free or a direct sum of cyclic p-groups). One can strengthen the notion of fully inert subgroup by defining H to be uniformly fully inert if there exists a positive integer n such that |(H + φH)/H| ≤ n for every φ ∈ End(G). The aim of this paper is to study the uniformly fully inert subgroups of abelian groups. A natural question arising in this investigation is whether such a subgroup is commensurable with a fully invariant subgroup. This paper provides a positive answer to this question for groups belonging to several classes of abelian groups.


Introduction
Unless otherwise stated, all groups considered in this paper are assumed to be abelian. For unexplained notions and notation we refer to the recent monograph by Laszlo Fuchs [21].

. Levels of invariance in the lattice of subgroups
If ϕ is an endomorphism of a group G and H a subgroup of G, then H is said to be ϕ-invariant if ϕH ≤ H. A subgroup F of a group G is fully invariant if it is ϕ-invariant for every endomorphism ϕ of G. Fully invariant subgroups have been an important tool in the theory of abelian groups; they form a complete sublattice of the lattice L(G) of all subgroups of G, which is denoted by Inv(G).
It may happen that the group G has only the two trivial fully invariant subgroups (this occurs precisely when G is a simple module over the ring End(G)). This makes it natural to replace the somewhat too restrictive relation of invariance of a subgroup with respect to an endomorphism by a weaker condition which is trivially satis ed also by all nite subgroups and all subgroups of nite index.

De nition 1.1. Let ϕ be an endomorphism and H a subgroup of a group G. Then H is called ϕ-inert if ϕ(H) ∩ H has nite index in ϕ(H)
passing to Jp-modules, in case of a torsion-free Jp-module which is either free or complete. In other words, the equality Inv˜(G) = I(G) holds for these groups. Thus, a fortiori, the equality Inv˜(G) = Iu(G) holds as well. These facts are the main results in the three papers [19], [24] and [25], respectively.
On the other hand, an example of a separable p-group G which admits a subgroup H ∈ I(G) \ Inv˜(G) is provided in [24]. Furthermore, an example of a torsion-free Jp-module A admitting a submodule K ∈ I(A) \ Inv˜(A) is provided in [25]. A careful analysis of the above examples shows that the subgroup H and the submodule K are not even uniformly fully inert (see Section 6). Therefore we have the more precise strict No examples are up to now available ensuring the strict inclusions, thus providing further support to Conjecture 1.6.
The strict inclusion Inv˜(G) I(G) is provided also by divisible groups G whose torsion-free rank rk (G) is nite and non-zero. Moreover, Inv˜(G) I(G) is proved by a recent result by Chekhlov [6], who characterized the completely decomposable groups G of nite rank satisfying this strict inclusion (see §3.2 for more details).
The main goal of this paper is to study the uniformly fully inert subgroups of abelian groups and to present old and new results related to Conjecture 1. 6.
In Section 2 we present some preliminary basic results and introduce the fully invariant hull and the fully invariant core of a subgroup, which are useful tools in our investigation.
In Section 3 we consider the chain (*) for divisible groups and completely decomposable groups recalling some known facts from [8] and [6,7], respectively.
In Section 4 we introduce three classes of groups which are of interest for our conjecture. The rst one is the class of Orsatti groups, which are the groups compact in the natural topology and whose structure was determined by Orsatti in [28]. The Orsatti groups are algebraically compact groups satisfying certain niteness conditions. The second class consists of groups with nite ranks, i.e., groups of nite torsion-free rank and nite p-ranks for all primes p. The third larger class contains the Orsatti groups and the groups with nite ranks and consists of the narrow groups G, de ned by the property that G/nG is nite for all positive integers n (De nition 4.1). These groups have been characterized in several ways in [17,Theorem 3.3], where it was proved that the adjoint algebraic entropy of any endomorphism of a narrow group is zero (see [17,Proposition 3.7]). The narrow groups are exactly the totally bounded groups in the natural topology.
In Section 5 we prove that our Conjecture 1.6 holds for groups with nite ranks (see Proposition 5.4) and for Orsatti groups (see Theorem 5.8).
As mentioned above, in Section 6 we give detailed proofs of the facts that the examples for the strict inclusions Inv˜(G) I(G), for G a p-group, and Inv˜(A) I(A), for A a Jp-module, are not even uniformly fully inert. Section 7 summarizes the state of art concerned with the conjecture and collects some open questions related to this conjecture.
Finally, we remark that many of the notions that we introduce in this paper, as well as some statements and proofs, can be carried out in the non-abelian context, see Remark 2.15 for more details.

Notation and terminology
For an abelian group G we denote by d(G) the maximal divisible subgroup, by t(G) the maximal torsion subgroup, by tp(G) the p-primary component of t (G). The order of an element g ∈ G is denoted by o(g). We follow [21,Chapter 3,Section 4] in de ning the rank rk(G) of a group G as the cardinal number of a maximal linearly independent system containing only elements of in nite and prime power orders. The cardinality of the subset of elements of in nite order is denoted by rk (G) and is called the torsion-free rank, while the cardinality of the subset of elements of order a power of the prime p is denoted by rkp(G) and is called the p-rank of G.
It is well known that rk (G) = dim Q (G ⊗ Q) and that rkp(G) is the dimension of the socle G[p] of tp(G) as vector space over Z/pZ. We will say that G is a group with nite ranks when rk (G) < ∞ and rkp(G) < ∞ hold for all primes p (obviously, this does not imply rk(G) < ∞).
If G = i G i is a direct sum, for each i we will denote by π i the projection π i : G → G i .

Preliminary results
We start this section by proving the fact mentioned in the Introduction, namely that a subgroup commensurable with a fully invariant subgroup is uniformly fully inert; we will freely use the following well-known facts holding for a group G: if G ≤ G ≤ G with G G < ∞, then for any X ≤ G and any ϕ ∈ End(G), both ϕ(G ) Proof. Consider the chain We have that the rst factor ϕ(K) ϕ(K)∩ϕ(H) of (2.1) has order which divides | K H∩K | since it is an epic image of ϕ(K) ϕ (K∩H) which in turn has order dividing | K H∩K |.

. Uniformly fully inert subgroups of nite direct sums
Let us see how the study of (uniformly) fully inert subgroups of nite direct sums can be reduced to box-like subgroups. Recall that a subgroup H of an arbitrary direct sum In general, we denote G (i) = j≠ i G j for each i ∈ I. Then, for any H ≤ G, π i (H) = (H + G (i) ) ∩ G i is the image of the canonical projection of H in G i . Moreover H i = H ∩ G i = H ∩ π i (H). It follows that any subgroup H is sandwiched between two box-like subgroups: Using this notation, we have the following lemmas. We omit the easy proof of the rst one, since it is similar to the proof of Lemma 2.4, which is in turn inspired by [

In particular, under such a condition, for each i the subgroup H i is uniformly fully inert in G i and ib(H i ) divides ib(H) .
Proof. We may assume n > and point out that -with some harmless abuse of notation-each ϕ ∈ End(G) can be written as where H (j) = i≠ j H i . Conversely, for each ϕ ∈ End(G) in the above notation we have The proof follows now easily. In particular, the last statement follows from the fact that To prove that h ∈ H, let ξ ∈ End(G ) be the inverse of n · id G ∈ End(G ); extend ξ to an endomorphism α of G by sending G to zero. As h = nα(h) ∈ nα(H) ⊆ H, we are done.
(2 b ) Since n·id G is invertible in End(G ) and it belongs to the center of that ring, the fact that nϕ(H ) ≤ H for every endomorphism ϕ of G implies that H is a fully invariant subgroup of G .
(2c) Assume that H ∈ Inv˜(G ), so there exists a fully invariant subgroup L ≤ G such that H ∼ L (H is commensurable with L). By our assumption on G and items (2a) and (2 b ), it follows that the subgroup L ⊕ H of G is fully invariant. Since obviously H = H ⊕ H ∼ L ⊕ H , we are done.

. The fully invariant hull and the fully invariant core
Every subgroup H of a group G is sandwiched between the minimal fully invariant subgroup H * of G  Proof. The inclusions Inv(G) ⊆ Inv * (G) and Inv(G) ⊆ Inv * (G) are obvious. Note that, if H if a subgroup of a group G with either |H/H * | < ∞ or |H * /H| < ∞, then H ∈ Inv˜(G). Furthermore, if H, K ∈ Inv * (G), then we have: and, dually, writing + in place of ∩ we have So Inv * (G) is a sublattice of Inv˜(G). The proof that Inv * (G) is a sublattice of Inv˜(G) is obtained in the same way as above, by writing H * , K * in place of H, K and H, K in place of H * , K * .
The next lemma will be used in the sequel (see Lemma 5.1).

Lemma 2.10. If H is a uniformly fully inert subgroup of the group G with inertial bound ib(H) = n, then
Hence, both H * /H and H/H * are bounded by n.
Proof. For each ϕ ∈ End(G), all subgroups ϕH have order dividing n modulo H, hence H * /H is bounded by n and consequently nH * ≤ H. Also nH * is fully invariant and nH * ≤ H, hence nH * ≤ H * . Thus nH * ≤ H and nH ≤ H * , so also H/H * is bounded by n.
Two natural questions concerning the sublattices Inv * (G) and Inv * (G) related to the sublattice Iu(G) arise.
Similar questions can be posed replacing Iu(G) by Inv˜(G) and I(G). Our interest in these question stems from the simple observation that a positive answer to any of these two items would imply that H ∈ Inv˜(G), according to Lemma 2.9, i.e., our main conjecture is "locally" solved, as far as the speci c subgroup H is concerned.
In case the answer is positive for all subgroups H of G, then G satis es our main conjecture.
We will see in Section 5 that the answer to Question 2.11 is positive for some subgroups H of G for a large classe of abelian groups G. In particular, it is true for all subgroups of torsion-free groups G of nite rank. However, in general the answer is in the negative, as the next examples show.
According to Lemma 2.9, Inv * (G) ∪ Inv * (G) ⊆ Inv˜(G). The next example shows that this inclusion is proper as well.
of G which is a fully invariant subgroup of G. Therefore, H ∈ Inv˜(G), so H is uniformly fully inert in G, by Corollary 2.2. We show now that H * = G and H * = , so that both H * /H = G/H and H/H * = H are in nite, i.e., the uniformly fully inert subgroup H simultaneously fails to satisfy both (I) and (II) in Question 2.11. To see that H * = G note that f can be sent by appropriate endomorphisms of G to any other element of G; so H * = G. To check that H * = assume, by way of contradiction, that ≠ x ∈ H * . If x has order p , it generates a cyclic summand, so it can be sent to e ∉ H (or to f ∉ H) by an appropriate endomorphism of G, contradicting the full invariance of H * and the inclusion H * ≤ H. If x has order p, either it has height , in which case it generates a cyclic summand, so it can be sent to e ∉ H, or x = py. In the latter case y generates a cyclic summand and can be sent to f by an appropriate endomorphism ϕ of G. Hence, ϕ(x) = ϕ(py) = pf ∉ H, again a contradiction.
As mentioned above, the subgroup H in Example 2.13 is commensurable with a fully invariant subgroup of G (namely, G[p]). More precisely, it was proved in [24] that every fully inert subgroup of a direct sum of cyclic p-groups has this property. Actually, the proof of this fact in [24] produces in the general case a fully invariant commensurable subgroup smaller than H * or H * . Another similar example is given in [24,Example 3.11].
Let us see now a further example which has an impact in the non-abelian context. Example 2.14. Let G = i∈Z a i with a i Z(p ) and H = a ⊕ i≠ , pa i . Then clearly H is commensurable with G[p] = pG = i∈Z pa i the socle of G. Let ϕ be the automorphism of G such that ϕ(a i ) = a i+ for each i (Bernoulli shift). Then, for all j, we have a j = ϕ j (a ) ∈ ϕ j (H) ≤ H * hence H * = G.
Therefore, H * /H is in nite and the same occurs to H/H * . In fact, since ϕ is invertible, we must have H * = ϕ j (H * ) for all j and clearly ϕ j (H * ) ≤ ϕ j (H). The elements in ϕ j (H) have (j + )-component equal . Thus Finally, the subgroup G − = −i∈N a i is ϕ-inert but not uniformly fully inert, as In the language of non-commutative groups, the argument of Example 2.14 proves exactly that in the wreath product Z(p ) Z G ϕ a subgroup may be commensurable with a normal subgroup even if it is not commensurable neither with its normal closure nor with its normal core. Moreover, the subgroup G − is inert in G (in Belyaev's terminology) but not uniformly inert (for a similar example see also [5], where a quotient of this group is considered). (b) The paper [4] provides an example of a nilpotent p-group G with the property that each subgroup of G is commensurable with a normal subgroup even if there are subgroups H of G which do not have the property that |H : H G | is nite and also subgroups H which do not have the property that |H G : H| is nite. Here, as usual, H G (resp. H G ) denotes the normal core (resp. normal closure) of H in G.

Known facts for divisible groups and completely decomposable groups . The chain (*) for divisible groups
We consider now the chain (*) of sublattices of L(G) when G is a divisible group. Fully inert subgroups of divisible group have been studied in [16] and some more details have been furnished in [8]. From [8,Proposition 5.6] and [16,Theorem 4.9] we get the following Example 3.1. For the torsion-free divisible group of nite rank Q n the following holds: and I(Q n ) consists of the homogeneous completely decomposable groups of rank n, i.e., of those subgroups of the form H ⊕ . . . ⊕ Hn, where the H i 's are all isomorphic to a xed subgroup of Q.
More generally, for divisible groups the picture is the following.

. The chain (*) for completely decomposable groups
A completely decomposable group is a torsion-free group which is isomorphic to a direct sum of rational groups, i.e., of subgroups of the rational group Q. We refer to [21, Chapter 12, Section 1] for the basic notions of characteristic and type and for their properties, and for the fundamental fact proved by Baer that the types are in bijection with the isomorphism classes of the rational groups, thus providing their classi cation.
Here we recall just few things. A characteristic is a sequence of non-negative integers or symbols ∞ indexed by the prime numbers Two characteristics are declared equivalent if they have the same ∞'s and nitely many di erences in the other entries. A type is the equivalence class of a characteristic with respect to this equivalence relation (our notation is slightly di erent from that in [21], where di erent brackets are used): Given two characteristics χ = (kp)p and χ = (hp)p and the two corresponding types t and t , their products are de ned by where ∞+n = ∞ and ∞+∞ = ∞. The inequality χ ≥ χ is de ned pointwise; accordingly, t ≥ t for the types means that the same inequality holds for two suitable characteristics representing them. With these orders the characteristics form a complete distributive lattice, while the types form a non-complete distributive lattice (e.g., the sequence of types tn = [n, n, . . . , n, . . .] (n ≥ ) has no supremum). If t ≥ t and χ ≥ χ represent these types, the colon of the two types is given by where ∞ − n = ∞ and ∞ − ∞ = ∞.
. .] is idempotent if, for every p, either hp = or hp = ∞. The reduced type t of the type t is obtained from t replacing all the hp's di erent from ∞ by ; obviously t = t : t.
If G is a torsion-free group, the characteristic (or height sequence) of an element a ∈ G is the sequence of the p-heights of a: where hp(a) is the maximal k ∈ N such that a ∈ p k G, if such a k does exist, or ∞ otherwise. The type t(a) of a is the type represented by χ(a).
The type t(A) of a rational group A is, by de nition, the type t(a) of any ≠ a ∈ A. If A ≤ B are rational groups, then t(A) ≤ t(B) and t (A) ≤ t (B).

Given two rational groups A and B, Hom(A, B) is naturally isomorphic to a rational group and t(A) t(B) implies Hom(A, B) = , while t(A) ≤ t(B) implies t(Hom(A, B)) = t(A) : t(B).
For every rational group A of type t, the endomorphism ring Hom (A, A) is isomorphic to A , the rational group of reduced type t , and Hom(A , A)) ∼ = A. From this fact it is easy to deduce the behavior of the chain (*) for rational groups. We include detailed proofs as a prelude to the next results on completely decomposable groups. In particular, Proposition 3.3 should be compared with Corollary 3.5. Proof. Recall that an endomorphism ϕ of G is induced by the multiplication by a rational number m/n with (m, n) = , where n is a product of primes p such that G = pG, since Hom(G, G) ∼ = G as rings.
(2) First we note that t (G) = t (H) precisely when pG = G implies pH = H, as the inequality t (H) ≤ t (G) is always true.
If t (H) ≠ t (G), i.e., pG = G and pH < H for some prime p, then the endomorphism ϕ of G induced by the multiplication by /p does not satisfy ϕH ≤ H, so H is not fully invariant. Conversely, assume that pH = H for all p such that pG = G and let ϕ : G → G be as above. The equality nH = H clearly implies that ϕH ≤ H, so H is fully invariant.
(3) Assume that H ∈ Iu(G). According to (2), to prove that H ∈ Inv(G) we have to check that t (G) = t (H). Assume for a contradiction that pG = G and pH < H for some prime p. Then the endomorphism ψn of G induced by the multiplication by /p n satis es (H + ψn H)/H ∼ = H/p n H, whose cardinality tends to ∞.
(4) From (1) and (2)  In the sequel we address the class of completely decomposable groups. For general information on these groups we refer to [21, Chapter 12, Section 3].
The following theorem concerning the chain (*) for completely decomposable groups of nite rank combines results from three papers, namely [6][7][8]  It follows from this theorem that a uniformly fully inert subgroup H of a completely decomposable groups of nite rank has nite index in its fully invariant hull H * . In Corollary 5.3 we will obtain the notably stronger result that this holds for an arbitrary torsion-free group of nite rank.
Let us recall also the concluding result for homogeneous completely decomposable groups G = A ⊕ . . . ⊕ An of nite rank, de ned by the property that the subgroups A i are all isomorphic; their common type is then the type t(G) of G. Recently Chekhlov extended in [7] Theorem 3.4 to completely decomposable groups of arbitrary rank, but with only a nite number of homogeneous components (i.e., the maximal homogeneous summands). First he showed that it is enough to consider reduced completely decomposable groups, and for these groups he proved the following

e. every fully inert subgroup of G is commensurable with a fully invariant subgroup; 2. if a homogeneous component G i has nite rank, then: (a) t(G i ) contains no ∞;
(b) if t(G i ) < t(G j ) for some j, then rk (G j ) ≥ ℵ ; (c) under the hypothesis of (b), if there are no types t(G k ) between t(G i ) and t(G j ), the type t(G j ) : t(G i ) is idempotent.
We conjecture that one may add to the equivalent conditions of Theorem 3.6 also the equality I(G) = Iu(G), i.e., every fully inert subgroup is uniformly fully inert (see Question 7.4 at the end of the paper).

Orsatti groups, groups with nite ranks and narrow groups
In this section we introduce three classes of abelian groups which are strictly related to each other, namely, the Orsatti groups introduced in [28], the narrow groups de ned in [15], and the groups with nite ranks already de ned in the Introduction.
In the sequel we make use of a functorial topology, the so called natural topology, or Z-topology, according to some authors (see for more details and background [21,27]). It is de ned as follows: for an abelian group G the natural topology topology ν G has as basic neighborhoods of 0 the subgroups nG = {ng | g ∈ G}, running n over the set of all positive integers. Clearly, ν G is Hausdor precisely when the rst Ulm subgroup G = n> nG of G is trivial. It is easy to see that this is equivalent to asking G to be residually nite [14].
Orsatti investigated in [28] the class of abelian groups G such that (G, ν G ) is compact. These groups are named Orsatti groups in [12,20], where the class of these groups is denoted by O. The description of Orsatti groups G, obtained in [28], is as follows: where Jp denotes the compact group of p-adic integers, kp ∈ N and Fp is a nite abelian p-group. In other words, G is a direct product of nitely generated Jp-modules, for p ∈ P. It is immediate to verify that an Orsatti group G satis es the property that G/nG is nite for every integer n. The Orsatti groups have also the following remarkable property established in [28] under the assumption of the Generalized Continuum Hypothesis: they are precisely the abelian groups G that admit a unique compact group topology, namely ν G . Since ν G is functorial, every ϕ ∈ End(G) is automatically continuous.
Inspired by Orsatti's de nition of the class O, one can consider the class of abelian groups G satisfying the property that G/nG is nite for every integer n; in topological terms, (G, ν G ) is totally bounded. It is well known that for residually nite groups this amounts to say that the completion of (G, ν G ) is compact. These groups are named according to the following De nition 4.1. [15] An abelian group G is said to be narrow whenever G/nG is nite for every natural number n > . We denote by N the class of all narrow groups.
The class N of narrow groups was introduced in [15] as an example of a large class of abelian groups G such that every endomorphism of G has adjoint algebraic entropy zero, and was also highlighted in [14]. The adjoint algebraic entropy is the natural dual of the algebraic entropy for endomorphisms of abelian groups, deeply investigated in [13,18] (see also the survey [23]). In [15] the following characterizations of narrow groups are presented, involving the family C(G) of all subgroups of nite index of G.

Theorem 4.2. [15, Theorem 3.3] For an abelian group G the following conditions are equivalent: (a) G is narrow; (b) G/pG is nite for every prime number p; (c) the family C(G) is countable; (d) |C(G)| < ℵ ; (e) C(G) contains a countable decreasing co nal chain; (f) the natural topology ν G of G coincides with the pro nite topology of G.
In the next theorem we provide some more properties of the narrow groups. In particular, it gives various reduction to relevant classes of groups, as reduced groups, torsion and torsion-free groups. (b) If D is a divisible group, then D = nD for all n ∈ Z, hence trivially D ∈ N. This makes the nal assertion obvious, as the class N is closed under taking nite direct sums and quotients, according to item (a). To see that an Orsatti group belong to N it is enough to consider the local case, which immediately follows from the fact that Jp /p k Jp is nite for all k ≥ .
(c) is trivial in view of (a), as G/nG ∼ = (G/G )/(nG/G ) for all n ∈ Z.
(d) follows from item (a), taking into account that t(G) is pure in G.
(e) Write G = tp(G), the primary decomposition of G. For each n ∈ Z, G/nG is a quotient of a direct sum of nitely many tp(G)'s. Thus, using (a), we deduce that G ∈ N if and only if tp(G) ∈ N for all primes p. So we can assume that G is a reduced p-group for some prime p. Pick a basic subgroup B of G. If G ∈ N, then G/pG ∼ = B/pB is nite, henceforth B, as a direct sum of cyclic p-groups, must be nite, and consequently G = B is nite. The converse is trivial.
(f) Making use of item (d), we can consider separately the cases when G is torsion-free and when G is torsion. Moreover, since the class of groups of nite rank is closed under taking subgroups, we can assume, in view of (b), that the group G is reduced. If G is a torsion-free group of nte rank, then the fact that G ∈ N (i.e., G/nG is nite for each n > ) is proved in [1, Theorem 0.1]. The torsion case follows directly from item (e), since a reduced p-group of nite rank is nite.
(g) Suppose that G ∈ N is a reduced torsion-free group. Thus (G, ν G ) is a Hausdor totally bounded group. Then its completion K is a compact group. Moreover, for every n > the closure of the ν G -open subgroup nG coincides with nK (the density of nG in nK is obvious, so it su ces to note that nK is compact, hence closed, as a continuous image of the compact group K under the multiplication by n). Since nG is closed in G, one has nG = G ∩ nG = G ∩ nK, this proves that G is a pure subgroup of K.
Since, by the de nition of completion, the group K has as basic neighbourhoods of 0 the nite-index subgroups nK = nG, this implies that (K, ν K ) is compact. Now Orsatti's theorem [28] applies to provide a topological isomorphism of K with a group of the form p J kp p , where the kp's are non-negative integers. Conversely, a pure subgroup of a group of this form belongs to N, by items (a) and (b).
Item (g) of the above theorem shows that if a torsion-free reduced group G belongs to N, then |G| ≤ ℵ . On the other hand, all non-trivial groups of the form G = p J kp p obviously belong to N and are torsion-free, reduced and of size ℵ .
The class N of narrow groups is not closed under taking arbitrary subgroups; for instance, Jp ∈ N, but a free subgroup of Jp of in nite rank does not belong to N (obviously, a free abelian group belongs to N if and only if has nite rank). The next corollary determines the hereditary core of N, namely, the largest subclass of N closed under taking arbitrary subgroups.

Corollary 4.4. All subgroups of a group G belong to N if and only if G is a group with nite ranks.
Proof. By the de nition of rank, it is obvious that the class of groups with nite ranks is closed under taking subgroups, and item (b) in Theorem 4.3 ensures that these groups are narrow. So, assuming that all subgroups of a group G belong to N, we must prove that rk (G) and rkp(G) are nite for all primes p. The divisible part d(G) is a group with nite ranks, otherwise it contains either a free group of in nite rank, or an elementary p-group of in nite dimension, which fail to be narrow. Hence we can assume that G is reduced. By item (d), it is enough to separately check t(G) ∈ N and G/t(G) ∈ N. Item (e) gives the conclusion for t(G), while if G is torsion-free it must have nite rank, by the same argument used for divisible groups.
We denote the subclass of N consisting of the groups with nite ranks by FR. We have seen that the following inclusion holds: Note that a countable group G belongs to O if and only it is nite, since the in nite compact groups have size at least ℵ . Hence, the above inclusion is strict, witnessed by the group ℵ Q (or any other countably in nite divisible group of in nite torsion-free rank). The intersection FR ∩ O consists of the nite groups.

Positive answers to Conjecture 1.6
In this section we prove that Conjecture 1.6 holds for groups with nite ranks and for Orsatti groups, i.e., for groups in the class FR ∪ O considered above. We start with a result regarding uniformly fully inert subgroups which are narrow. 2. By Lemma 2.10, the quotient H * /H is bounded by n, and nH * , as a subgroup of H, has also nite rank. Since H * ∼ = nH * , H * has nite rank, so it is narrow. By the same argument as above, H * /H is nite. Proof. Let H be a uniformly fully inert subgroup of G ∈ FR. Then H ∈ FR as well, according to Corollary 4.4. Now Lemma 5.1 implies that H/H * is nite, i.e., H ∈ Inv * (G). By Lemma 2.9, this proves that Inv˜(G) = Iu(G) = Inv * (G).
The next example shows the existence of a torsion-free group G of in nite rank with Inv˜(G) = Iu(G) that is neither free nor divisible. For a discussion on the issue of whether one can add to the chain of equalities (5.1) also I(Jp), see Questions 7.8 and 7.9. The preceding Example 5.5 is the starting point to prove the following results concerning local Orsatti groups. By a local Orsatti group we mean an Orsatti group which is a ( nitely generated) Jp-module for some p.
Proposition 5.6. If G is a local Orsatti group, then every uniformly fully inert subgroup is either nite or of nite index, therefore Inv˜(G) = Iu(G).
Proof. First we prove the claim in the torsion-free case. In [24] it is proved that a non-zero submodule H of a free Jp-module A of nite rank r belongs to I(G) if and only if H has nite index in A. Here we must assume that H is just a non-zero subgroup, and not a submodule, of the free Jp-module A, but we have the stronger assumption that H is uniformly fully inert; we want to prove that still H has nite index in A. The case r = is covered by Example 5.5. Let A = ≤i≤r A i , where r > and A i ∼ = Jp for all i. Let π i : A → A i ≤ A be the canonical projections. Now we prove that H ∩ A i ≠ for all i. This is ensured, by Corollary 2.7(b) applied to ψ = π i , for all i with π i (H) ≠ (as π i (H) ≠ implies that π i (H) is in nite, since A i ∼ = Jp is torsion-free). Since H ≠ { }, there exists an index j such that π j (H) ≠ . For every index i x an isomorphism ϕ i : A j → A i ; then ψ i = ϕ i · π j is a homomorphism A → A i such that π i (H) ≠ , so π i (H) is in nite, hence Lemma 2.7(b) can be applied for ψ = ψ i to entail H ∩ A i ≠ .
From Corollary 2.7(a) and Example 5.5 we derive that H ∩ A i = p n i A i , with n i > for all i ≤ r, so H ≥ ≤i≤r p n i A i . The last group is of nite index in A, so such is H.
Let now G = A ⊕ F be a local Orsatti group, where A is as above and F is a nite p-group, and let H be a non-zero uniformly fully inert subgroup of G. If H ∩ A = , then H embeds in F, so it is nite. Otherwise, H ∩ A ≠ has nite index in A, by Corollary 2.7 (a) and what proved above, therefore G/H, as an epic image of (A/A ∩ H) ⊕ (F/F ∩ H), is nite. Thus we conclude that H is either nite, or of nite index in G, as desired, hence Inv˜(G) = Iu(G).
We are now in the position to prove the main result of this section. Then G = G ⊕ G and each G i is a fully invariant subgroup of G. Since n is invertible modulo p ∈ P , we can apply Proposition 2.8 and derive, using its notation, that H = H ⊕ H with H a fully invariant subgroup of G . In order to conclude that H ∈ Inv˜(G), according to item (2c) in Proposition 2.8, it is su cient to check that H ∈ Inv˜(G ).
For every p ∈ P , let πp : G → Gp be the projection. Applying again Proposition 2.8 to the subgroup H of G , we deduce that each πp(H ) is a uniformly fully inert subgroup of Gp for all p ∈ P . By Proposition 5.6, we know that πp(H) is either nite, or of nite index in Gp. The hypothesis that H is uniformly fully inert in G, by using the projections πp : G → Gp (p ∈ P ), yields: Being P a nite set, we obtain that H ∈ Inv˜(G ).

When I u (G) I(G): fully inert but not uniformly fully inert subgroups
The goal of this section is to prove that the examples of p-group G and torsion-free Jp-module X, exhibited in [24] and [25] respectively, which demonstrate the strict inclusions Recall that a basic subgroup B of a p-group G is called semi-standard if B = n≥ Bn, with Bn either zero or isomorphic to the direct sum of fn copies of Z(p n ) for some positive integers fn. The integers fn coincide with the so-called Ulm-Kaplansky invariants (of nite index) of B, and the direct summands Bn are called the homogeneous components of B.
In [24,Theorem 4.2] it is proved that, given a reduced separable p-group G of cardinality ℵ with semistandard basic subgroup B, and with endomorphism ring End(G) = Jp · G ⊕ Es(G), where Es(G) is the twosided ideal of the small endomorphisms (see [21,Chapter 7, Section 3] for its de nition), the socle B[p] of the basic subgroup B is in nite and fully inert, but it fails to be commensurable with a fully invariant subgroup of G. The crucial point in the proof is that B is countable and any fully invariant non-trivial subgroup of G has cardinality ℵ .
Thus the proper inclusion Inv˜(G) I(G) is proved for such a group G, and it is natural to ask whether the subgroup B[p] belongs to Iu(G), i.e., whether it is uniformly fully inert. In order to prove that this fails to be true, we need the following lemma. To conclude, extend the restriction ξ := η B ⊕···⊕Bn : B ⊕ · · · ⊕ Bn → G of the isomorphism (6.5) to an endomorphism ϕ of G making use of the well-known fact that B ⊕ · · · ⊕ Bn is a direct summand of G, so ξ can be extended by sending to a complement of B ⊕ · · · ⊕ Bn. Then ϕ is the desired endomorphism, as ϕ B ⊕···⊕Bn = ξ is injective and ϕ(B) ≤ L trivially meets B.
We can now prove that in Theorem 4.
Therefore, Lemma 6.1 apples to produce an endomorphism ϕ of G such that its restriction to B ⊕ · · · ⊕ Bn is injective, and ϕ(B ⊕ · · · ⊕ Bn) ∩ B = . Then ϕ satis es the desired inequality |( The p-groups satisfying the requirements of Proposition 6.2 are the so-called Pierce-like groups; they were rst constructed in [29,Theorem 15.4]. The natural problem arises: does the equality Inv˜(G) = Iu(G) hold true for the p-group G in Proposition 6.2?
We pass now to torsion-free Jp-modules. It is well known that the non-zero fully invariant submodules of a torsion-free reduced Jp-module M are the obvious ones, that is, those of the form p k M for some non-negative k (see [26,Exercise 72(b)]).
In [25] it is proved that there exists a torsion-free Jp-module X containing a fully inert submodule H which is non-commensurable with p k X, for all non-negative k. The Jp-module X, furnished by a realization theorem of commutative rings as endomorphism rings of Jp-modules (see [22,Theorem 4.1]), has its endomorphism ring satisfying End(X) = Jp · X ⊕ E (X), where E (X) is the two-sided ideal of the endomorphisms with nite rank image. The required submodule H is any H ≤ X such that pX < H and both X/H and H/pX are in nite. Thus the proper inclusion Inv˜(X) I(X) is proved for such a Jp-module X.
We prove that the submodule H does not belong to Iu(X). Proof. In order to prove that H / ∈ Iu(X), it is enough to nd for every k ∈ N an endomorphism ϕ k of X, such that |(H + ϕ k H)/H| ≥ k. (6.6) Fix a k ∈ N and let π : X → X/pX be the canonical homomorphism. The subspace π(H) = H/pX of the vector space X/pX splits, let X/pX = H/pX ⊕ K/pX, where K is a Jp-submodule of X containing pX. Our hypotheses on H imply that both H/pX and K/pX are in nite-dimensional. Let H = {h i : i ∈ N} and K = {k i : i ∈ N} be two sequences in H and K, respectively, such that the sets {π(h i )} i∈N and {π(k j )} j∈N are linearly independent in the vector spaces H/pX and K/pX.
Choose n ∈ N with p n+ ≥ k and let H be the Jp-submodule of X generated by h , h , . . . hn. By the choice of H, the submodule H is pure and free, so H = h Jp ⊕ · · · ⊕ hnJp. As Jp, hence H as well, is pure-injective, this yields that H is a direct summand of X. Therefore, the map sending h , h , . . . , hn, respectively, to the elements k , k , . . . , kn of K can be extended to an endomorphism ϕ k of X sending the complement of H to zero. By the choice of K,
The next consequence of Proposition 6.3 shows that also for the module X, which provides an example for the strict inequality Inv˜(X) I(X), the equality Inv˜(X) = Iu(X) of Conjecture 1.6 holds true. Theorem 6.4. Let X be the Jp-module of Proposition 6.3. Then any non-zero uniformly fully inert submodule K of X is commensurable with a submodule of the form p k X for some non-negative k. Therefore Inv˜(X) = Iu(X).
Proof. Let us note that we can replace K, if necessary, by a subgroup of K which is commensurable with K.
Since K is uniformly fully inert the quotient K * /K is bounded, hence, for a suitable m > , In [25] it is proved that every non-zero fully invariant submodule of X is of the form p k X for some k ≥ , hence there exist a minimal positive integer s and a maximal r > s such that p s X < K < p r X.
Let d := r − s. If d = , then we intend to apply Proposition 6.3 to deduce that K ∈ Inv˜(X). To this end we note that K is a uniformly fully inert submodule of p r X as well since every endomorphism of p r X (uniquely) extends to an endomorphism of X. Hence, Proposition 6.3 applied to p r+ X < K < p r X implies that either p r X/K or K/p r+ X are nite, hence K ∈ Inv˜(p r X). As p r X is fully invariant in X, this yields K ∈ Inv˜(X) as well.
Now assume d > and the assertion true for all submodules K of X with Using the fact that we can replace K by a subgroup of it which is commensurable with K, we can assume, without loss of generality, that (K + p r+ X)/p r+ X is in nite. Indeed, let K := K ∩ p r+ X and assume that (K + p r+ X)/p r+ X ∼ = K/K is nite. Then K ∼ K and p s X < K < p r+ X. As s − (r + ) < d, K ∈ Inv˜(X), by our inductive hypothesis. Hence, K ∈ Inv˜(X) as well.
From now on we assume that (K + p r+ X)/p r+ X is in nite.
Let us see that if p r X/(K + p r+ X) is nite, then K ∈ Inv(X). Indeed, in such a case p r X = p r+ X + K + F for some nitely generated subgroup F of X. Multiplying s − r − times by p and replacing one gets p r X = p s X + K + F = K + F, as K ≥ p s X. Since F is nitely generated, we deduce that p r X/K, being isomorphic (as an abelian group) to a quotient of F, is countable. As p r X/K is a nitely generated Jp-module and |Jp| = c, this yields that p r X/K is nite. This implies that K ∈ Inv˜(X).
From now on we assume that p r X/(K + p r+ X) is in nite and we shall see that this leads to a contradiction with our assumption that K is uniformly fully inert. Our next step is adapting the proof of Proposition 6.3 to obtain, for each positive integer n, an endomorphism ϕ of X such that |(K + ϕK)/K| ≥ n. To this end choose elements p r x , . . . , p r xn ∈ K which are independent modulo p r+ K, using the fact that (K + p r+ X)/p r+ X is in nite. The elements x , . . . , xn generate a direct summand H of X. Now use the fact that p r X/(K + p r+ X) is in nite, so there exist independent elements p r y + p r+ X, . . . , p r yn + p r+ X ∈ p r X/p r+ X in the complement of (K + p r+ X)/p r+ X. The assignment which sends each element x i to y i , for ≤ i ≤ n, can be extended to an endomorphism ϕ of X which sends the complement of H to zero; then the desired inequality |(K + ϕK)/K| ≥ n holds.

Conclusion
In this section we summarize the results concerning Conjecture 1.6 on uniformly fully inert subgroups obtained up to now, and we list a series of open questions.

. The state of art
For several classes of groups G the equality Inv˜(G) = I(G) holds, hence, a fortiori, also the equality Inv˜(G) = Iu(G) of Conjecture 1.6 holds. The list of these classes includes: -free groups (see [19]); -direct sums of cyclic p-groups (see [24]); -divisible groups D such that r (D) is either zero or ∞ (see Theorem 3.2); -completely decomposable groups of nite rank G such that their rank one summands have types which are either equal or incomparable and contain no ∞'s (see Theorem 3.4).
Furthermore, for divisible groups and completely decomposable groups of nite rank not satisfying the conditions above, we know that the equality of Conjecture 1.6 still holds.
On the other hand, the known examples of groups G such that the strict inequality Inv˜(G) I(G) holds, do not furnish counter-example to Conjecture 1.6 (see Section 6).
Finally, there are groups G for which we only know that the equality of Conjecture 1.6 holds. These groups are: -the torsion-free groups of nite rank (see Proposition 5.4); -the Orsatti groups (see Theorem 5.6).

. Open questions
For further investigation of Conjecture 1.6 we propose the following questions.
The rst series of questions concerns Theorem 5.7 asserting that an Orsatti group G satis es our main conjecture, i.e., every uniformly fully inert subgroup of G is commensurable with a fully invariant subgroup. The next question concerns the class of completely decomposable groups. A positive answer was conjectured after Theorem 3.6.
Question 7.4. Are the equivalent conditions of Theorem 3.6 equivalent also to the equality I(G) = I u (G), i.e., every fully inert subgroup is uniformly fully inert?
The following question was posed after Proposition 6.2.
Question 7.5. Does the equality Inv˜(G) = Iu(G) hold true for the p-group G in Proposition 6.2?
It is probably hopeless to prove Conjecture 1.6 for all abelian groups. A more reasonable strategy is to test various reasonably chosen classes of groups. To answer (c) it su ces to consider the problem for all p-groups, when p ranges over all primes.