A real chain condition for groups

We consider a very weak chain condition for a poset, that is the absence of subsets which are order isomorphic to the set of real numbers in their natural ordering; we study generalised radical groups in which this finiteness condition is set on the poset of subgroups which do not have certain properties which are generalizations of normality. This completes many previous results which considered (apparently) stronger chain conditions.


Introduction
The study of groups G with some finiteness condition on the poset of subgroups of G with (or without) given properties has been object of many investigations; see for example [9] for a survey up to 2009 and [4,5,6,7,8,11,12,17,27] for more recent contributions.The most elegant and popular ones have been perhaps those concerning the restrictions on the chains -i.e.totally ordered subsets -of χ (or non-χ) subgroups.Here, as throughout the paper, the letter χ denotes a property pertaining to subgroups.
Recall that a poset P of subgroups a group is said to have the weak maximal condition (Max-∞), the weak minimal condition (Min-∞), the weak double chain condition (DCC-∞), if P does not contain chains whose order type is the same of the natural numbers N, as the negative integers −N, as the whole set of integers Z respectively and, furthermore, in which each subgroup has infinite index in the successive (see [28]).When P is the poset of all χ (or non-χ) subgroups of a group G, it is usually said that G has the weak maximal condition on χ (non-χ, respectively) subgroups whenever P has Max-∞.The corresponding terminology is used for Min-∞ and DCC-∞.
It is clear that Max-∞, as well as Min-∞, imply DCC-∞, and many investigations have been carried out in order to find for which poset P of subgroups (or, equivalentely, for which property χ) these conditions are all equivalent.However, due to the existence of the so-called Tarski groups (i.e.infinite simple groups whose proper non-trivial subgroups have prime order), investigations in this area are usually completed under a suitable (generalised) solubility condition.Zaicev [28] proved that a locally (soluble-by-finite) group has DCC-∞ (on all subgroups) if and only if it has Max-∞ or Min-∞; moreover, any locally (soluble-by-finite) group with DCC-∞ is a soluble-by-finite minimax group.Recall that a group is said to be minimax if it has a finite series whose factor satisfy either the minimal or the maximal condition.
We say that a poset of subgroups of a group has the real chain condition (RCC) if it does not contain chains whose order type is the same as the set of real numbers R with their usual ordering.Note that DCC-∞ implies RCC (see Proposition 2.1, below).Thus, we are interested in finding for which properties χ the following holds.
Framework Statement Let χ be a property pertaining to subgroups.For a generalised radical group, the following are equivalent: (i) the weak minimal condition on non-χ subgroups; (ii) the weak maximal condition on non-χ subgroups; (iii) the weak double chain condition on non-χ subgroups.
(iv) the real chain condition on non-χ subgroups; Recall that a group is said generalised radical when it has an ascending (normal) series with locally (nilpotent or finite) factors.
In the following sections, it will be shown that the above Framework Statement holds when χ is one the properties: n=normal; an=almost normal; nn=nearly normal; m=modular; per=permutable.Recall that for each of the previous choiches for the property χ and for a generalised radical group G, the equivalence between the first three condition in the above Framework Statement is already known and it is also known that, with the exception of χ = an or χ = nn, each of these three items is equivalent to the property that either G is a soluble-by-finite minimax group or all subgroups of G have χ (see [3,6,7,14]).We will also prove the Framework Statement holds when χ is the property for a subgroup to be subnormal (but only in periodic soluble case).We will give futher references, detailed statements and proofs for the different properties in the following corresponding sections.For undefined terminology and basic results we refer to [22,23].
Until now, the most general chain condition in the theory of infinite groups appears to be the condition that the poset of non-χ subgroups has the (set theoretical) deviation (see [11,12,17,27]).We do not recall here the (recursive) definition of the deviation since, for our purposes, it is enough to recall the fact that a poset has deviation if and only if it contains no sub-poset order isomorphic to the poset D of all dyadic rationals m/2 n in the interval from 0 to 1 (see [19], 6.1.3).Since D is a countable dense poset without endpoints, it is order-isomorphic to the rational numbers by Cantor's isomorphism theorem.Therefore a poset has deviation if and only if it contains no sub-poset order isomorphic to the poset Q of rational numbers in their usual ordering.Clearly, any poset with deviation has RCC and the converse holds when the poset is complete but it does not seem always to be the case; moreover, any poset of subgroups with DCC-∞ has deviation and so also RCC (see Proposition 2.1, below).Therefore as a consequence of our Framework Statement, the equivalence between RCC and deviation is proved true for the poset of all non-χ subgroups of a group for our selection of properties χ.

Preliminary results
For brevity, we call Z-, Q-or R-chain a poset with the same order type as Z, Q or R respectively.Proposition 2.1 Let P be a poset of subgroups of a group G. Then (i) If P has DCC-∞, then P has deviation.
(ii) If P has deviation, then P has RCC.
(iii) If P is complete and has RCC, then P has deviation.
Proof.(i) Assume that P has not deviation, so it contains a strictly increasing family (X i ) i∈Q of subgroups, then (X i ) i∈Z is a Z-chain in which for each i ∈ Z the index |X i+1 : X i | is infinite because there are infinitely many subgroups X j with i < j < i + 1 and j ∈ Q. Hence P has not DCC-∞.
(ii) is trivial.(iii) Assume that P does not contain R-chains and, for a contradiction, let (X i ) i∈Q be a Q-chain of elements of P .Since P is complete, if Y r = sup{X i : i ≤ r} for every r ∈ R, we obtain that (Y i ) i∈R is an R-chain of P , a contradiction.
Since the poset of all subgroups of a group is complete, it follows by Proposition 2.1 that a group has RCC for all subgroups if and only if it has deviation.On the other hand, any direct product of infinite non-identity groups does contain an R-chain as we state in next elementary lemma.Lemma 2.2 If a group G is the direct product of infinitely many non-trivial subgroups, then G has not RCC.
Proof.Clearly G contains the direct product of countably many non-trivial subgroups and hence a subgroup of the form Dr i∈Q G i .Then, an R-chain is formed by the subgroups Tushev proved that a soluble group has deviation if and only it is minimax (see [27], Lemma 4.4).This result can be extended as in Theorem 2.4 below.Before, let us state a standard general result that will be used in what follows to reduce our investigation to radical-by-finite groups, where radical means that the group has an ascending (normal) series whose factors are locally nilpotent.Proposition 2.3 Let X be a class of groups which closed with respect to forming subgroups and homomorphic images, such that each locally finite X-group is soluble-by-finite.Then any generalised radical X-group G is radical-by-finite.
Proof.Let K a subgroup of G which is maximal with respect to be radical and normal in G, and let T /K be a subgroup of G/K which is maximal with respect to be locally finite and normal in G/K.By the properties of X we have that there is a normal subgroup S of T containing K such that S/K is soluble and T /S is finite.Then S G /K is soluble and so, by the maximality of K, we have that S G = K.Hence T /K is finite and T is radical-by-finite.
Let H/T be any normal subgroup of G/T which is locally nilpotent, then C H/K (T /K) is a normal locally nilpotent subgroup of G/K.Since G/K has no non-trivial locally nilpotent normal subgroups, we have that C H/K (T /K) is trivial; hence H/K is finite.It follows that H = T .Therefore G/T has no non-trivial subgroups which are either locally nilpotent or locally finite; on the other hand, G is generalised radical and hence G = T is radical-by-finite.

Theorem 2.4 Let G be a generalised radical group. Then the poset of all subgroups of G has RCC if and only if G is a soluble-by-finite minimax group.
Proof.If G is minimax, it is extension of groups with either the minimal or the maximal condition and hence G certainly has RCC, as the property RCC is closed under extensions as a standard argument shows.
Conversely, notice first that in any group with RCC, each abelian subgroup is minimax by Proposition 2.1 and the already quoted result by Tushev (see [27], Lemma 4.4).In particular, any locally finite group with RCC is a Chernikov group by a celebrated result by Shunkov [25] (and hence is soluble-by-finite).Therefore if G is a generalised radical group with RCC, then G is radical-by-finite by Proposition 2.3 and so it is a soluble-byfinite minimax group (see [22] Part 2, Theorem 10.35).
Let us state now a technical general key lemma which will be useful later.
The next result applies when χ is the property of being a normal subgroup or, more generally, when χ is the property of being Γ -invariant for some subgroup Γ of the automorphism group of the group.It also holds when χ is one of the the properties an, nn (see [20], Lemma 1), sn (see [23], 13.14 and 13.1.5);moreover, item (i) holds also for the properties m and per (see [24], pp.201-202).
Lemma 2.6 Let G be a group with RCC on non-χ subgroups.If G contains a section H/K which is the direct product of an infinite collection of non-trivial subgroups, then the following hold: Proof.Write H/K = H 1 /K ×H 2 /K where both H 1 /K and H 2 /K are the direct product of an infinite collection of non-trivial subgroups.Application of Lemma 2.5 yields that there exist an Let G be a group with RCC on non-χ subgroups, where χ is such that the intersection X ∩ Y is a χ-subgroup whenever X and Y are χ-subgroups.Let L be any subgroup of G.If there exists a subgroup H of G which is the direct product of an infinite collection of L-invariant non-trivial subgroups and such that Proof.Write H = H 1 × H 2 where both H 1 and H 2 are the direct product of an infinite collection of non-trivial subgroups.Application of Lemma 2.5 yields that there exist subgroups Finally, we state as a lemma a property of abelian groups which is probabily wellknown and that we will use in our aurgument without further mention.For a proof of such a property see, for instance, Lemma 3.2 of [11].
Lemma 2.8 Any abelian group which is not minimax has an homomorphic image which is the direct product of infinitely many non-trivial subgroups.

Real chain condition on non-normal subgroups
Let G be a group.The F C-centre of G is the subgroup consisting of all elements having finitely many conjugates, and G is said to be an F C-group if it coincides with its F Ccentre.The class of all F C-groups have been widely studied.It tourns out, in particular, that if G is an F C-group then G/Z(G) and G ′ are locally finite (see [26], Theorem 1.4 and Theorem 1.6), moreover G/Z(G) is residually finite (see [26], Theorem 1.9).
A subgroup H of G is called nearly normal if the index |H G : H| is finite.Any group whose (cyclic) subgroups are nearly normal is an F C-group (see [26], Lemma 7.12), moreover all (abelian) subgroups of a group are nearly normal if and only if the group is finite-by-abelian (see [26], Theorem 7.17).
A subgroup H of G is called almost normal if it has finitely many conjugates in G, i.e. when the index |G : N G (H)| is finite.Clearly, if all (cyclic) subgroups of G are almost normal then G is an F C-group; moreover, all (abelian) subgroups of a group are almost normal if and only if the group is central-by-finite (see [26], Theorem 7.20).Notice that any central-by-finite group is finite-by-abelian (see [26], Theorem 1.2).

Lemma 3.1 Let G be a group with RCC on non-(almost normal) (resp. non-(nearly
Proof.The factor G/Z(G) is periodic and so we may consider a torsion-free subgroup [26], Lemma 1.3) and hence A is both nearly normal and almost normal in this case.Suppose now that A is not a Chernikov group, hence A has an homomorphic image which is the direct product of infinitely many non-trivial subgroups and hence A is almost normal in G (resp.nearly normal) by Lemma 2.6.Therefore all abelian subgroups of G are almost normal (resp.nearly normal) and so lemma follows by above quoted results.
Next three lemmas allows us to assume that abelian subgroups have finite total rank; where the total rank of an abelian group is the sum of all p-ranks for p = 0 or p prime.Recall also that a well-know result of Kulikov states that any subgroup of a direct product of cyclic subgroups is likewise a direct product of cyclic subgroups (see [10], Theorem 3.5.7), in what follows we make use of this result also without further reference.

Lemma 3.2 Let G be a group with RCC on non-(almost normal) (resp. non-(nearly normal) subgroups, and let A be a subgroup which is the direct product of infinitely many non-trivial cyclic subgroups. Then all subgroups of A are almost normal (resp. nearly normal) subgroups of G.
Proof.Let X be any cyclic direct factor of A. Clearly we may write A = X × A 1 where A 1 is not finitely generated, and so application of Lemma 2.7 gives that X is almost normal (resp.nearly normal) in G. Therefore A is contained in the F C-centre of G and all finitely generated subgroup of A are almost normal (and nearly normal) in G. On the other hand, if Y is any subgroup of A which is not finitely generated, then Y is likewise a direct product of cyclic subgroups, and hence Y is almost normal (resp.nearly normal) in G by Lemma 2.6.Therefore all subgroups of A are almost normal (resp.nearly normal) in G. Lemma 3.3 Let G be a group and let A be a normal subgroup of G which is the direct product of infinitely many non-trivial cyclic subgroups.If all subgroups of A are almost normal in G, then A contains a subgroup which is the direct produt of infinitely many finitely generated G-invariant non-trivial subgroups.
Proof.Let A 1 = {1} and assume that G-invariant subgroups A 1 , . . ., A n of A have been constucted in such a way that A 1 , . . ., A n = A 1 × • • • × A n is finitely generated.Then there exists subgroups X and Y such that Y is finitely generated, A 1 , . . ., A n ≤ Y and A = X × Y .Since X has finitely many conjugates in G, the factor A/X G is finitely generated; in particular, X G is not trivial and so we may choose a non-trivial element Proof.We will prove firstly that G contains a subgroup which is a direct product of infinitely many non-trivial normal subgroups.
Let A be a subgroup of G which is is the direct product of infinitely many non-trivial cyclic subgroups.By Lemma 2.7 follows easily that every cyclic subgroup of A is almost normal in G, hence A is contained in the F C-centre F of G. Since F/Z(F ) is finite by Lemma 3.1, we may clearly suppose that A ≤ Z(F ).Let T be the subgroup consisting of all elements of finite order of Z(F ), and assume first that T is not a Chernikov group.Since T is the direct product of its primary components, which are normal subgroups of G, in order to prove our claim it can be assumed that π(T ) is finite.Then there exists a prime p such that the Sylow p-subgroup P of T does not satisfy the minimal condition, so that the socle of P is an infinite abelian normal subgroup of G of prime exponent and hence application of Lemma 3.2 and Lemma 3.3 give us the required subgroup.Assume now that T is a Chernikov group, so that Z(F ) has infinite torsion-free rank.Let U be a free subgroup of Z(F ) such that Z(F )/U is periodic; in particular, U has infinite rank.Then U is almost normal in G by Lemma 2.6, so that also Z(F )/U G is periodic.Thus U G ≃ U is a free abelian normal subgroup of infinite rank of G and again application of Lemma 3.2 and Lemma 3.3 prove that G contains the claimed subgroup.
Therefore G contains a subgroups which is a direct product of infinitely many nontrivial normal subgroups.Then it follows from Lemma 2.7 that all cyclic subgroups are almost normal, so that G is an F C-group and application of Lemma 3.1 concludes the proof.

Lemma 3.5 Let G be a radical-by-finite group with RCC on non-(almost normal) subgroups. Then each non-minimax subgroup of G is almost normal.
Proof.Let H be any non-minimax subgroup of G, then H contains an abelian nonminimax subgroup A (see [22] Part 2, Theorem 10.35).Let B any free subgroup of A such that A/B is periodic.If B is not finitely generated, then G/Z(G) is finite by Lemma 3.4 and so H is almost normal.Thus assume that B is finitely generated, so that A/B does not satisfy the minimal condition and hence its socle is infinite.Thus B is almost normal by Lemma 2.6, so that also the periodic group A/B G has infinite socle and hence G/B G is finite over its centre by Lemma 3.4.Since any central-by-finite group is also finite-by-abelian, it follows that G ′ is polycyclic-by-finite. Thus the abelian factor H/H ′ is not minimax and so it has an homomorphic image which is the direct product of infinitely many non-trivial subgroups; hence H is almost normal in G by Lemma 2.6.

Lemma 3.6 Let G be a locally finite group with RCC on non-(almost normal) subgroups.
Then either G is a Chernikov group or G/Z(G) is finite.In particular, G is abelian-byfinite.
Proof.Assume that G is not a Chernikov group.Then G contains an abelian subgroup A which does not satisfy the minimal condition (see [25]); thus the socle of A is a direct product of infinitely many non-trivial groups of prime order and hence G/Z(G) is finite by Lemma 3.4.
It has been proved in [3] that for a generalised radical group, weak minimal, weak maximal and weak double chain condition on non-(almost normal) subgroups are equivalent, moreover, a description of generalised radical groups statisfying such a condition is also given in the case of groups which are neither minimax nor central-by-finite.Now we are in position to prove our Framework Statement when χ = an, it add another equivalent condition to the weak chain conditions (and so also to the deviation) on non-(almost normal) subgroups.

Theorem 3.7 Let G be a generalised radical group. Then the following are equivalent: (i) G satisfies the weak minimal condition on non-(almost normal) subgroups; (ii) G satisfies the weak maximal condition on non-(almost normal) subgroups; (iii) G satisfies the weak double condition on non-(almost normal) subgroups. (iv) G satisfies the real chain condition on non-(almost normal) subgroups.
Proof.As already quoted, conditions (i), (ii) and (iii) are equivalent, and imply (iv) by Proposition 2.1.Conversely, if G satisfies (iv), then G is radical-by-finite by Lemma 3.6 and Proposition 2.3, so that Lemma 3.5 yields that each non-minimax subgroup of G is almost normal and hence Theorem 12 of [3] can be applied to conclude the proof.
We turn to consider the case when χ = nn.First step is to restrict the total rank of abelian subgroups.

Lemma 3.8 Let G be a group with RCC on non-(nearly normal) subgroups. If G has a subgroup which is the direct product of infinitely many non-trivial cyclic subgroups, then G ′ is finite.
Proof.Let A be a subgroup of G which is is the direct product of infinitely many nontrivial cyclic subgroups, then A is a nearly normal subgroup of G by Lemma 2.6.Since it is well-know that any abelian-by-finite group has a characteristic abelian subgroup of finite index, it follows that A G contains a G-invariant abelian subgroup N of finite index.Clearly, A ∩ N has finite index also in N so that N is likewise a direct product of infinitely many non-trivial cyclic subgroups (see [10], Theorem 3.5.7 and Exercise 8 p.99).Replacing A with N it can be supposed that A is a normal subgroup of G.Moreover, all subgroups of A are nearly normal subgroups of G by Lemma 3.2.
Let T be the subgroup consisting of all elements of finite order of A. Then T is normal in G and T is the direct product of non-trivial cyclic subgroups by Kulikov's Theorem already quoted; moreover, all subgroups of A/T are normal in G/T (see [2], Lemma 2.7).If T is finite, it follows easily from Lemma 2.7 that every cyclic subgroup of G/T is nearly normal; hence G/T is an F C-group and application of Lemma 3.1 yields that G ′ is finite.
Therefore it can be assumed that A = T is infinite.Then A contains a G-invariant subgroup D which is a finite-by-divisible such that all subgroups of A/D are normal in G/D (see [2], Theorem 2.11).Since A is the direct product of non-trivial cyclic subgroups, also D is likewise the direct product of non-trivial cyclic subgroups.Hence D must be finite and so, as before, it can be obtained that G ′ is finite.

Lemma 3.9 Let G be a locally finite group with RCC on non-(nearly normal) subgroups.
Then either G is a Chernikov group or G ′ is finite.In particular, G is soluble-by-finite.
Proof.Assume that G is not a Chernikov group.Then G contains an abelian subgroup A which does not satisfy the minimal condition (see [25]); thus the socle of A is a direct product of infinitely many non-trivial groups of prime order and hence G ′ is finite by Lemma 3.8.
In [6], it has been proved that for a generalised radical group, weak minimal, weak maximal and weak double chain condition on non-(nearly normal) subgroups are equivalent; moreover, with the exception of finite-by-abelian groups, it tourns out that for generalised radical groups, weak chain conditions on non-(nearly normal) subgroups are equivalent to weak chain conditions on non-(almost normal) subgroups.In next result we prove that Framework Statement holds when χ = nn so that, in particular, real chain condition is equivalent to the weak chain conditions (and so also to the deviation) for such a subgroup property.

Theorem 3.10 Let G be a generalised radical group. Then the following are equivalent: (i) G satisfies the weak minimal condition on non-(nearly normal) subgroups; (ii) G satisfies the weak maximal condition on non-(nearly normal) subgroups; (iii) G satisfies the weak double condition on non-(nearly normal) subgroups. (iv) G satisfies the real chain condition on non-(nearly normal) subgroups.
Proof.Since conditions (i), (ii) and (iii) are equivalent (see [6], Theorem A) and imply (iv) by Proposition 2.1, it is enough to prove that (iv) implies (iii).Let G satisfy RCC on non-(nearly normal) subgroups.Lemma 3.9 and Proposition 2.3 give that G is radical-by finite.By Lemma 3.1 it can be assumed that G is not an F C-group so that G does not contain subgroups which are a direct product of infinitely many non-trivial cyclic subgroups by Lemma 3.8.Hence all abelian subgroups have finite total rank and so G has a subgroup of finite index having a finite series in which each factor is abelian of finite total rank (see [1]).It follows that G has finite (Prüfer) rank and so each nearly normal subgroup is also almost normal (see [13], Lemma 3.1).Therefore G has RCC on non-(almost normal) subgroups, so that G satisfy the weak double chain condition on non-(almost normal) subgroups by Theorem 3.7 and thus also the weak double chain condition on non-(nearly normal) subgroups (see [6], Theorem 2.12).
Groups in which all subgroups are normal are well-known since a long time and are the well described Dedekind groups (see [23], 5.3.7).Moreover, Kurdachenko and Goretskiȋ [14] showed that for locally (soluble-by-finite) groups, the weak minimal condition on nonnormal subgroups is equivalent to the weak maximal condition on non-normal subgroups, and any locally (soluble-by-finite) group satisfying such a condition is either a solubleby-finite minimax group or a Dedekind group (in particular, these result remains true for generalised radical groups by Proposition 2.3).We extend this result to condition RCC and improve Corollary 1 of [12] which handles only the periodic case.(ii) G satisfies the weak maximal condition on non-normal subgroups; (iii) G satisfies the weak double condition on non-normal subgroups.
(iv) G satisfies the real chain condition on non-normal subgroups.
(v) either G is a soluble-by-finite minimax group or all subgroups of G are normal.
Assume first that G contains a subgroup A which is the product of countable many nontrivial cyclic subgroups; then G ′ is finite by Lemma 3.8.Let X be any cyclic subgroup of G. Since |G : C G (X)| is finite, replacing A with a suitable subgroup, it can be assumed that [A, X] = A∩X = {1} and hence application of Lemma 2.7 to the subgroup X, A = A×X gives that X is normal in G.It follows that G is a Dedekind group in this case.
Therefore it can be assumed that G does not contain subgroups which are a direct product of infinitely many non-trivial cyclic subgroups; hence all abelian subgroups have finite total rank.Therefore the soluble radical of G has a finite series in which each factor is abelian of finite total rank (see [1]); in particular G has finite (Prüfer) rank.Suppose that G is not a minimax group, so that it contains an abelian non-minimax subgroup (see [22] Part 2, Theorem 10.35).Hence either G ′ is finite or G/Z(G) is polycyclic-by-finite (see [6], Lemma 2.7).Therefore G ′ is polycyclic-by-finite (see [22] Part 1, p.115).Let H be any non-minimax subgroup of G. Then the abelian factor H/H ′ is not minimax and so it has an homomorphic image which is the direct product of infinitely many non-trivial subgroups; thus H is normal in G by Lemma 2.6.Therefore all non-minimax subgroups of G are normal and hence G certainly has the weak minimal condition on non-normal subgroups.

Real chain condition on non-modular subgroups
A subgroup H of a group G is said to be modular if it is a modular element of the lattice of all subgroups of G, i.e., if Lattices in which all elements are modular are called modular.Clearly every normal subgroup is modular, but modular subgroups need not be normal; moreover, a projectivity (i.e., an isomorphism between subgroup lattices) maps any normal subgroup onto a modular subgroup, thus modularity may be considered as a lattice generalization of normality.A subgroup H of a group G is said to be permutable (or quasinormal) if HK = KH for every subgroup K of G; and the group G is called quasihamiltonian if all its subgroups are permutable.It is well-known that a subgroup is permutable if and only if it is modular and ascendant, and that any modular subgroup of a locally nilpotent group is always permutable (see [24], Theorem 6.2.10).Groups with modular subgroup lattice, as well as quasihamiltonian groups, have been completely described and we refer to [24] as a general reference on (modular) subgroup lattice.In particular, recall that every non-periodic group with modular subgroup lattice is quasihamiltonian, and that a periodic group is quasihamiltonian if and only if it is a locally nilpotent group in which every subgroup is modular.Moreover, any group with modular subgroup lattice is metabelian provided it is non-periodic or locally finite.
Recently, in [7], weak chain conditions on non-modular subgroups have been studied.It tourns out that for a generalised radical group, weak minimal and weak maximal condition on non-modular subgroups are both equivalent to the property that all nonminimax subgroups are modular and characterizes groups which either are soluble-byfinite and minimax or have modular subgroup lattice.Here we complete the description by considering RCC.Lemma 4.1 Let G be a group with RCC on non-modular subgroups having section H/K which is a direct product of infinitely many non-trivial subgroups.If x is an element of G such that x ∩ H ≤ K, then there exists a subgroup L of H such that both L and x, L are modular subgroup of G.
Proof.As already noted, conditions (i), (ii) and (v) are equivalent, and clearly imply (iii); moreover, (iii) implies (iv) by Proposition 2.1.On the other hand, since any locally finite group with modular subgroup lattice is soluble (see [24], Theorem 2.4.21), if G satisfies (iv), then G is radical-by-finite by Lemma 4.2 and Proposition 2.3 and so it satisfies (v) by Lemma 4.3.The theorem is proved.
In [7], weak chain conditions on non-permutable subgroups have been also considered and it was proved that all results on weak chain conditions on non-modular subgroups have a corresponding with non-permutable subgroups.Here the wished results for groups in which the poset of all non-permutable subgroups has RCC can be obtained just replacing modular subgroups with permutable subgroups in the above arguments or, in an independent way, as a consequence of the following.Lemma 4.5 Let G be a periodic locally soluble group with RCC on non-permutable subgroups.Then either G is a Chernikov group or all subgroups of G are permutable.
Proof.Assume that G is not a Chernikov group and let x, y ∈ G. Clearly x, y is finite and so, since G is locally soluble, there exists an abelian x, y -invariant subgroup A which does not satisfy the minimal condition (see [29]).Replacing A by its socle, it can be assumed that A is the direct product of infinitely many cyclic groups of prime order.Application of Lemma 3.3 gives that A contains a subgroup B which is the direct product of infinitely many non-trivial finite x, y -invariant subgroups.Clearly it can be assumed that B ∩ x, y = {1}, hence Lemma 2.5 yields that B contains a normal subgroup B * such that x B * = B * x is permutable in B. Hence and so, since B ∩ x, y = {1}, it follows that x y ⊆ y x .Similarly y x ⊆ x y and hence x y = y x .Therefore all (cyclic) subgroups of G are permutable.(ii) G satisfies the weak maximal condition on non-permutable subgroups; (iii) G satisfies the weak double condition on non-permutable subgroups.
(iv) G satisfies the real chain condition on non-permutable subgroups.
(v) either G is a soluble-by-finite minimax group or all subgroups of G are permutable.
Proof.As in Theorem 4.4, it is enough to prove that (iv) implies (v).Hence assume that G satisfies (iv).Theorem 4.4 yields that either G is a soluble-by-finite minimax groups or has modular subgroup lattice, so that application of Theorem 2.4.11 of [24] and Lemma 4.5 give that (v) holds, and so the theorem is proved.

Real chain condition on non-subnormal subgroups
The weak minimal and the weak maximal condition on non-subnormal subgroups have been considered in [15] and in [16] respectively.It turns out that if G is a generalised radical group G satisfying the weak minimal condtions on non-subnormal subgroups, then either G is a soluble-by-finite minimax group or any subgroup of G is subnormal.On the other hand, there exists non-minimax groups satisfying the weak maximal condition on non-subnormal subgroups which still have non-subnormal subgroups.Indeed, if G = A ⋊ g where A = Dr i∈N a i is an infinite elementary abelian p-group (p prime) and g is the automorphism of infinite order of A such that [a 1 , g] = 1 and [a i+1 , g] = a i for all i ≥ 1, then G is an hypercentral non-minimax group satisfying the weak maximal condition on non-subnormal subgroups which is not a Baer group (see [16]).Recall here that the Baer radical of a group is the subgroups generated by all cyclic subnormal subgroups and a group is said to be a Baer group if it concides with its Baer radical; in particular, in a Baer group all finitely generated subgroups are subnormal and nilpotent.Notice that the above example G = A ⋊ g does not satisfy the weak minimal condition on non-subnormal subgroups but the poset of all non-subnormal subgroups of G has deviation (see the introduction of [17]), and so also RCC by Proposition 2.1.Hence Framework Statement cannot be proved in his form when χ = sn is the property for a subgroup to be subnormal .However, for locally finite groups the weak minimal condition on non-subnormal subgroups is equivalent to the weak maximal condtion on non-subnormal subgroups, and here we are able to prove the Framework Statement when χ = sn within the universe of periodic soluble groups, improving Theorem 1 of [12] which concernes with soluble periodic groups with deviation on the poset of non-subnormal subgroups.
Lemma 5.1 Let G be a periodic group with RCC on non-subnormal subgroups.If G contains an abelian subgroup A which does not statisfy the minimal condition, then G is a Baer group.
Proof.Replacing A by its socle it can be supposed that A is the direct product of infinitely many cyclic non-trivial subgroups.As a consequece of Lemma 2.7 it can be obtained that all cyclic subgroups of A are subnormal in G, hence A is contained in the Baer radical R of G and hence R does not satisfy the minimal condition.Let g be any element of G. Then R, g is locally soluble and hence there is no loss of generality if we assume that A is g -invariant (see [29]).Then A has finite index in A, g and hence all subgroups of A are almost normal in G. Thus Lemma 3.3 yelds that A contains a subgroup which is the direct produt of infinitely many finitely generated g -invariant non-trivial subgroups, and so it follows from Lemma 2.7 that g ∈ R. Thus G = R is a Baer group.Proof.This follows from [25] and Lemma 5.1.
In our argument we need the following easy remark.Lemma 5.3 Let G be a group and let N be a normal subgroup.If N satisfies maximal (resp.minimal) condition on G-invariant subgroups and G/N satisfies the weak maximal (resp.weak minimal) condition on normal subgroups, then G satisfies the weak maximal (resp.weak minimal) condition on normal subgrops.
Proof.Let (G i ) i∈N be an ascending chain of normal subgroups of G. Then (G i ∩ N ) i∈N is an ascending chain of G-invariant subgroups of N and hence there exists a positive integer n such that the index G i+1 ∩ N = G i ∩ N for any i ≥ n.On the other hand, (G i N/N ) i≥n is an ascending normal chain and so there exits a positive integer m ≥ n such that the index |G i+1 N/N : G i N/N | is finite for any i ≥ m.Then for every i ≥ m we have that the index is finite.Thus the result with weak maximal conditions is proved, the corresponding resut with weak minimal conditions can be proved similarly.
Recall that if G is a periodic Baer group, any subnormal abelian divisible subgroup is contained in the centre of G (see for instance [15], Lemma 5.1).
gives that there exists a positive integer j ≤ k such that A j X is not subnormal in A j−1 X.Since A j X ′ is contained in A j X and it is normal in A j−1 X, the factor group A j−1 X/A j X ′ is likewise a minimal counterexample.Thus we may replace G by A j−1 X/A j X ′ and X by A j X/A j X ′ , i.e. it can be supposed that X is abelian.
Since any abelian group which is not minimax has an homomorphic image which is the direct product of infinitely many non-trivial subgroups and since X is not subnormal in G, Lemma 2.6 gives that X is a Chernikov group.Let D be the largest divisible subgroup of X and let F be a finite subgroup, such that X = DF .Then [A, D] = {1} by Lemma 5.4, so that AD is a normal abelian subgroup of G = AX = (AD)F and hence G is nilpotent because F is finite and G is a Baer group.This contradiction completes the proof.
a finitely generated subgroup of A, and so lemma follows.Lemma 3.4  Let G be a group with RCC on non-(almost normal) subgroups.If G has a subgroup which is the direct product of infinitely many non-trivial cyclic subgroups, then G/Z(G) is finite.

Theorem 3 .
11 Let G be a generalised radical group.Then the following are equivalent:(i) G satisfies the weak minimal condition on non-normal subgroups;

Theorem 4 . 6
Let G be a generalised radical group.Then the following are equivalent:(i) G satisfies the weak minimal condition on non-permutable subgroups;

Corollary 5 . 2
Let G be a locally finite group RCC on non-subnormal subgroups.Then G is either a Chernikov group or a Baer group.