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Journal of 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on

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Francesco de Giovanni and Carmela Musella

Dipartimento di Matematica e Applicazioni, Uni¨ersita` di Napoli ‘‘Federico II’’, Complesso Uni¨ersitario Monte S. Angelo, Via Cintia, I 80126, Naples, Italy

and

Yaroslav P. Sysak1

Institute of , Ukrainian National Academy of Sciences, ¨ul. Tereshchenki¨ska 3, 01601 Kie¨, Ukraine

Communicated by Gernot Stroth

Received November 14, 2000

DEDICATED TO BERNHARD AMBERG ON THE OCCASION OF HIS 60TH BIRTHDAY

1. INTRODUCTION

A subgroup of a G is called modular if it is a modular element of the lattice ᑦŽ.G of all of G. It is clear that everynormal subgroup of a group is modular, but arbitrarymodular subgroups need not be normal; thus modularitymaybe considered as a lattice generalization of normality. Lattices with modular elements are also called modular. Abelian groups and the so-called Tarski groupsŽ i.e., infinite groups all of whose proper nontrivial subgroups have prime . are obvious examples of groups whose subgroup lattices are modular. The structure of groups with modular subgroup lattice has been described completelybyIwasawa wx4, 5 and Schmidt wx 13 . For a detailed account of results concerning modular subgroups of groups, we refer the reader towx 14 .

1 This work was done while the third author was visiting the Department of Mathematics of the Universityof Napoli ‘‘Federico II.’’ He thanks the G.N.S.A.G.A. of the Istituto Nazionale di Alta Matematica for financial support.

738 0021-8693r01 $35.00 Copyright ᮊ 2001 byAcademic Press All rights of reproduction in anyform reserved. ALMOST MODULAR SUBGROUP LATTICES 739

A subgroup H of a group G is said to be almost normal if the conjugacy Ž. class of H in G is finite or equivalentlyif the normalizer NHG of H has finite index in G. A wonderful theorem of Neumannwx 9 states that all subgroups of a group G are almost normal if and onlyif the centre ZGŽ. of G has finite index. If ␸ is an from the lattice ᑦŽ.G onto the subgroup lattice of a group G and N is a of G, then the N ␸ of N is a modular element of the lattice ᑦŽ.G . Further- more, ␸ maps everysubgroup of finite index of G to a subgroup of finite index of G Žseewx 14, Theorem 6.1.7. . Thus the image of anyalmost normal subgroup of G is modular in a subgroup of finite index of G. We shall saythat a subgroup H of a group G is almost modular if there exists a subgroup of finite index K of G containing H such that H is a modular element of the lattice ᑦŽ.K . The definition of almost modular elements can be given in an arbitrarylattice, and a lattice ᑦ will be called almost modular if all its elements are almost modularŽ see Section 2 for details. . Thus everylattice-isomorphic image of a group whose subgroups are almost normalŽ. i.e., of a central-by- is a group with almost modular subgroup lattice, and the aim of this article is to determine, having in mind results similar to Neumann’s theorem, the structure of such groups. Most of our paper is devoted to the studyof periodic groups with almost modular subgroup lattice, whose structure is completelydescribed bythe following theorem.

THEOREM A. Let G be a periodic group. The subgroup lattice ᑦŽ.Gis almost modular if and only if G s M = K, where M is a group with modular subgroup lattice, K is an abelian-by-finite group containing a finite normal subgroup N such that the lattice ᑦŽ.KrN is modular and ␲ Ž.Ž.M l ␲ K s л.

Moreover, it will also be proved that everyperiodic group G with almost modular subgroup lattice contains a normal subgroup of finite index L whose subgroups are modular in G Ž.see Theorem 5.11 in the last section , so that G looks like a central-by-finite group with L instead of ZGŽ.. It is well-known that a special role among modular subgroups is played bypermutable subgroups; here a subgroup H of a group G is said to be permutable Ž.or also quasi-normal if HK s KH for everysubgroup K of G. A group is called quasi-hamiltonian if all its subgroups are permutable. By a result of Stonehewerwx 15 , quasi-hamiltonian groups coincide with locally nilpotent groups having modular subgroup lattice. We shall saythat a subgroup H of a group G is almost permutable if it is permutable in a subgroup of finite index of G. The structure of groups in which every subgroup is almost permutable will be relevant for our purposes. 740 DE GIOVANNI, MUSELLA, AND SYSAK

THEOREM B. Let G be a periodic group. Then all subgroups of G are almost permutable if and only if G s Q = K, where Q is a quasi-hamiltonian group, K is an abelian-by-finite group containing a finite normal subgroup N such that KrN is quasi-hamiltonian and ␲ Ž.Q l ␲ Ž.K s л. As a consequence of Theorem B, it will be proved that all subgroups of a periodic group G are almost permutable if and onlyif G is simultane- ously finite-by-quasi-hamiltonian and quasi-hamiltonian-by-finiteŽ see Corollary5.9 in the last section .. Since central-by-finite groups can be characterized as groups which are both abelian-by-finite and finite-by- abelian, this result can be viewed as a natural extension of Neumann’s theorem. If G is anynon-periodic group with modular subgroup lattice, it was proved byIwasawa that the elements of finite order of G form a subgroup T and the factor group GrT is abelian; moreover, if G is not abelian, the group GrT is locallycyclic.The following theorem extends this result to nonperiodic groups whose is almost modular; it shows in particular that torsion-free groups with almost modular subgroup lat- tices are abelian. Recall here that the FC-centre of a group G is the subgroup consisting of all elements of G with finitelymanyconjugates, and a group is said to be an FC-group if it coincides with its FC-centre. Clearly, a group is an FC-group if and onlyif all its cyclicsubgroups are almost normal.

THEOREM C. Let G be a nonperiodic group with almost modular sub- group lattice. Then: Ž.a The set T of all elements of finite order of G is a normal subgroup, and the factor group GrT is abelian. Ž.b E¨ery subgroup of T is almost normal in G. Ž.c Either G is an FC-group or the group GrT is locally cyclic. It follows from the structure of groups with modular subgroup lattice that such groups are metabelian, provided theyhave no Tarski factor groups. As a consequence of Theorem A and Theorem C, we will prove the following result.

COROLLARY D. Let G be a group which has no Tarski homomorphic images. If the subgroup lattice ᑦŽ.G is almost modular, then G is metabelian- by-finite. Most of our notation is standard and can be found inwx 11 . We shall use the monographwx 14 as a general reference for results on subgroup lattices. We thank the referee for his useful comments and in particular for having shortened the original proof of Lemma 4.12. ALMOST MODULAR SUBGROUP LATTICES 741

2. SOME PRELIMINARIES

Let ᑦ be a lattice with least element 0 and greatest element I. Recall ᑦ ¨ that an element x of is co ered irreducibly byelements x1,..., xm of the interval wxxr0 if, for each element y of wxxr0 such that wxyr0isa with the maximal condition, there is i F m such that F Ä4 y xi, and the set x1,..., xm is minimal with respect to such a property. Clearlya subgroup H of a group G is covered irreduciblyin the lattice ᑦŽ. G byits subgroups H1,...,Hm if and onlyif H is the set-theoretic union of H1,...,Hm and none of these subgroups can be omitted from the covering. An element h of the lattice ᑦ is said to be cofinite if there exists a finite chain in ᑦ

s - - иии - s h h01h ht I

s y such that, for every i 0, 1, . . . , t 1, hi is a maximal element of the wxr lattice hiq1 0 and one of the following conditions is satisfied:

ⅷ h q is covered irreduciblybyfinitelymanyelements k ,...,k of i 1 1 n i ᑦ such that k n иии n k F h . 1 nii ⅷ ␸ wxr For everyautomorphism of the lattice hiq1 0 , the element n ␸ wxr wr n ␸ x hiih is modular in hiq1 0 and the lattice hiq1 hiih is finite. We shall saythat an element a of ᑦ is almost modular if there exists a cofinite element h of ᑦ such that a F h and a is a modular element of the lattice wxhr0 . The lattice ᑦ is called almost modular if all its elements are almost modular. A theorem of Schmidt yields that a subgroup H of a group G is cofinite in the lattice ᑦŽ.G if and onlyif H has finite index in G Žseew 14, Theorem 6.1.10x. . Therefore, a subgroup X of G is almost modular if and onlyif it is an almost modular element of the lattice ᑦŽ.G , and hence the subject of this article is the structure of groups with almost modular subgroup lattice. Our first lemma shows in particular that if G is a periodic group with almost modular subgroup lattice, then either G is locallyfinite or it must have a Tarski section.

LEMMA 2.1. Let G be a periodic group with almost modular subgroup lattice. If G is not locally finite, then there exists a subgroup H of G satisfying the following conditions: Ž.i H s ²x, M :, where ² x : and M are modular subgroups of H; Ž.ii M is a maximal and locally finite subgroup of H; Ž. r iii the factor group H MH is a Tarski group. 742 DE GIOVANNI, MUSELLA, AND SYSAK

Proof. Let L be a maximal locallyfinite subgroup of G, and let X be a subgroup of finite index of G such that L is a modular subgroup of X. Since G is not locallyfinite, L is properlycontained in X and there exists an element x of X _ L such that x p belongs to L for some prime number p. Since L is a modular subgroup of ²:x, L , the lattices w²:x, L rLx and w²:²:x r x l Lx are isomorphic, and so L is a of ²:x, L . Let H be a subgroup of finite index of ²:x, L such that ²:x is a modular subgroup of H, and put M s L l H. Then H s ²:x, M , and ²:x and M are modular subgroups of H. Moreover, the locallyfinite subgroup M is maximal in H because L is maximal in ²:x, L , and the index <

LEMMA 2.2. Let G s ²:x, y be a residually finite p-group, where p is a prime number. If²: x is a modular subgroup of G, then G is finite. Proof. Let G be anyfinite homomorphic image of G. Then ²:x is a permutable subgroup of G, and hence G s ²:²:xyhas bounded order. It follows that G is finite.

LEMMA 2.3. Let G be a residually finite p-group, where p is a prime number. If the lattice ᑦŽ.G is almost modular, then G is locally finite. Proof. Assume bycontradiction that G is not locallyfinite, so that by Lemma 2.1 it contains a subgroup H s ²:x, M , where M is locallyfinite, ²: r s ² : x is modular in H, and H MHHis a Tarski group. Then H Mx, y for some element y of H, and it follows from Lemma 2.2 that ²:x, y is finite, a contradiction. A normal subgroup N of a group G is said to be hypercyclically embedded in G if it has an ascending series with cyclic factors consisting of normal subgroups of G. It has been proved bySchmidt 14wx that, if H is a cyclic modular subgroup of a finite group G, the normal closure H G of H is hypercyclically embedded in G Žseewx 14, Theorem 5.2.5. .

LEMMA 2.4. Let G be a periodic residually finite group with almost modular subgroup lattice. Then G is locally finite. Proof. Assume bycontradiction that G is not locallyfinite, so that by Lemma 2.1 it contains a subgroup H s ²:x, M , where M is locallyfinite, ²: r s ² : x is modular in H, and H MHHis a Tarski group. Then H Mx, y for some element y of H. Put X s ²:x, y , and let X be anyfinite homomorphic image of X.As²:x X is hypercyclically embedded in X,it ALMOST MODULAR SUBGROUP LATTICES 743 follows that X is supersoluble, and so its commutator subgroup XЈ is nilpotent. Therefore the commutator subgroup XЈ of X is residually nilpotent, and hence it is the direct product of its Sylow subgroupsŽ seew 11, Part 2, Corollaryto Theorem 6.14x .. On the other hand, everySylow subgroup of XЈ is locallyfinite byLemma 2.3, so that XЈ is likewise locally finite and X is finite. This contradiction proves the lemma. A group G is called a P*-group if it is the of an abelian normal subgroup A of prime exponent bya cyclicgroup ²:x of prime-power order such that x induces on A a power automorphism of prime orderŽ recall here that a power automorphism of a group G is an automorphism mapping everysubgroup of G onto itself. . It is easyto see that the subgroup lattice of any P*-group is modular, and Iwasawawx 4, 5 proved that a locallyfinite group has modular subgroup lattice if and only if it is a direct product

s G Dr Gi , igI where each Gi is either a P*-group or a primarylocallyfinite group with modular subgroup lattice, and elements of different factors have coprime orders. Recall also that a group G is said to be a P-group if either it is abelian of prime exponent or G s ²:x h A is a P*-group with the subgroup ²:x of prime order. Let the group G s E = H be the direct product of a finite group E whose subgroup lattice is not modular, and an infinite P*-group H such that E and H have no elements of the same prime order. Then G is a locallyfinite group containing subgroups with infinitelymanyconjugates and the lattice ᑦŽ.G is not modular. On the other hand, if X is any subgroup of G, the intersection X l E is normal in XH, and

ᑦŽ.Ž.XHrX l E , ᑦ H is a , so that X is an almost modular subgroup of G. Therefore the lattice of subgroups of G is almost modular. This example shows that there exist locallyfinite groups with almost modular subgroup lattice that neither have modular subgroup lattice nor are central-by-finite. Recall that a subgroup H of a group G is said to be P-embedded in G if r G HG is a periodic group and the following conditions are satisfied:

ⅷ r s ŽŽr ..= r r G HGiDr g IiS H GL H G, where each S iH Gis a non- abelian P-group; ⅷ in the above direct decomposition, elements from different factors have coprime orders; 744 DE GIOVANNI, MUSELLA, AND SYSAK

ⅷ r s Ž Žr ..= ŽŽl .r . r H HGiDr g IiQ H GH L H G, where each Q iH Gis r a nonnormal Sylow subgroup of SiGH ; ⅷ H l L is a permutable subgroup of G. All P-embedded subgroups are modular, and it can be proved that every modular subgroup of a locallyfinite group is either permutable or P-em- beddedŽ seewx 17, Theorem 3.2 and Theorem E. .

LEMMA 2.5. Let G be a locally finite group, and let J be the finite residual of G. Then e¨ery modular subgroup of G contained in J is permutable in G. Proof. Assume bycontradiction that J contains a modular subgroup H of G which is not permutable in G, so that H is P-embedded in G Žsee wx14, Theorem 6.2.17. . Since every P-group is residuallyfinite, it follows from the definition that there exists a normal subgroup L of G such that GrL is residuallyfinite and H l L is permutable in G. Thus L contains J and H s H l L is a permutable subgroup of G. This contradiction proves the lemma.

LEMMA 2.6. Let ᑦ be a lattice with least element 0, and let a be an element of ᑦ such that x is modular in the inter¨alwx a k xr0 for each element x of ᑦ. Then a is a modular element of ᑦ. Proof. Let x and y be elements of ᑦ such that x F y, and put z s Ž.a k x n y. Then a n z s a n y. Since x is a modular element of wxa k xr0 , it follows that Ž.a k x n y s z s Ž.a k x n z s x k Ž.a n z s x k Ž.a n y . A similar argument shows that Ž.a k x n b s a k Ž.x n b for everyele- ment b of ᑦ such that a F b. Therefore a is a modular element of ᑦ. The finite residual of a group with almost modular subgroup lattice has the following useful property.

LEMMA 2.7. Let G be a group with almost modular subgroup lattice, and let J be the finite residual of G. Then e¨ery subgroup of J is modular in G. Moreo¨er, if G is locally finite, e¨ery subgroup of J is permutable in G. Proof. Let H be a subgroup of J, and let X be anysubgroup of G.As X is almost modular in G, there exists a subgroup of finite index Y of G such that X is a modular subgroup of Y. In particular, X is modular in the group ²:H, X , and it follows from Lemma 2.6 that H is a modular subgroup of G. Finally, if G is locallyfinite, the subgroup H is permutable in G byLemma 2.5. Recall that a group G is called an extended Tarski group if it contains a cyclic nontrivial normal subgroup N with prime-power order such that ALMOST MODULAR SUBGROUP LATTICES 745

GrN is a Tarski group and H F N or N F H for everysubgroup H of G. The existence of groups of this type has been proved by Ol’shanskiı˘ wx 10 . Clearlyanyextended Tarski group does not contain proper subgroups of finite index, and its subgroup lattice is modular. The last result of this section reduces the studyof periodic groups with almost modular subgroup lattice to the case of locallyfinite groups.

THEOREM 2.8. Let G be a periodic group. The subgroup lattice ᑦŽ.Gis almost modular if and only if G is a direct product of Tarski groups, extended Tarski groups, and a locally finite group with almost modular subgroup lattice such that elements of different factors ha¨e coprime orders. Proof. If G has the structure described in the statement, then the lattice ᑦŽ.G is isomorphic to the direct product of the subgroup lattices of the direct factors of G, and hence it is almost modular. Conversely, suppose that the periodic group G has almost modular subgroup lattice, and let J be the finite residual of G. Then all subgroups of J are modular in G byLemma 2.7, and in particular ᑦŽ.J is a modular lattice. Therefore J s T = L, where T is a direct product of Tarski groups and extended Tarski groups such that elements of different factors have coprime orders, and L is a locallyfinite subgroup with ␲ Ž.T l ␲ Ž.L s л Žseewx 14, Theorem 2.4.16. . Obviously T is a normal subgroup of G, and the factor group GrT is locallyfinite byLemma 2.4. Consider anyelement xT of GrT with prime order q, so that T is a maximal subgroup of X s Tx²:. Let Y be anysubgroup of X which is not contained in T, and let Z be a subgroup of finite index of X such that Y is a modular element of the lattice ᑦŽ.Z . Then X s TY and Z s YT Žl Z .s X,asT does not contain proper subgroups of finite index. Since all subgroups of T are modular in X, it follows that X has modular subgroup lattice, so that X s T = ²:g for some element g of order q and q does not belong to the set ␲ Ž.ŽT seewx 14, Theorem 2.4.16.Ž.Ž. . Therefore ␲ T l ␲ GrT s л, and all elements of G of order q centralize T.If a is anyelement of G with ny 1 order q n for some n G 2, the subgroup ²aq :²:is normal in T, a , and we obtain byinduction on n that

ny 1 wxT , a F T l ²aq : s Ä41, so that a acts triviallyon T. Let y and z be ␲ Ј-elements of G, where ␲ s ␲ Ž.T , so that ²y, z :is contained in the centralizer of T, and hence ²:y, z l T F ZT Ž.. As the factor group ²:y, zTrT is finite, it follows that also ²:y, z must be finite. Moreover, ²:y, z l T is a Hall normal ␲-sub- group of ²:y, z , and so there exists a ␲ Ј-subgroup E of ²:y, z such that ²:Ž²:y, z s y, z l TE .. The subgroup E centralizes T, so that E is normal in ²:y, z , and hence ²:y, z s E is a ␲ Ј-group. Therefore the set 746 DE GIOVANNI, MUSELLA, AND SYSAK

K of all ␲ Ј-elements of G is a locallyfinite subgroup, and G s T = K has the required structure. The theorem is proved.

3. NONPERIODIC GROUPS

In this section Theorem C will be proved. We first consider the be- haviour of elements of finite order in groups whose cyclic subgroups are almost modular.

LEMMA 3.1. Let G be a group, and let x and y be elements of finite order of G. If the subgroup²: x is modular in G, then xy and yx ha¨e finite order. Proof. As the subgroup ²:x is modular in G, the subgroup lattice ᑦŽ²xy :r ²xy :l ²x :. is isomorphic to the interval

wxwx²x, xy :²:r x s ²x, y :²:r x ,

and hence also to ᑦŽ²y :r ²x :l ²y :.. Thus ²xy :r ²xy :l ²x : is a finite group, and xy has finite order. It follows that also the element yx s Ž.xyy1 y1 y1 has finite order. Ž² :. LEMMA 3.2. Let G be a group, and let xiig I be a collection of finite s ²:N g cyclic modular subgroups of G. Then the subgroup X xi i Iis periodic. Proof. Everyelement x of X can be written in the form

␧ ␧ x s x 1 иии x t , ii1 t ␧ s " F where i1,...,itjbelong to I and 1 for all j t. We shall prove by induction on t that x has finite order; obviouslyit can be assumed that t ) 1 and that ␧ ␧ y s x 1 иии x ty1 ii1 ty1 ²: has finite order. Since xi is a modular subgroup of G, it follows from ␧ t Lemma 3.1 that x s yx t has finite order. i t Our next result shows in particular that, if G is a group with almost modular subgroup lattice, the elements of finite order of G form a subgroup.

THEOREM 3.3. Let G be a group whose cyclic subgroups are almost modular. Then the set of all elements of finite order of G is a subgroup. Proof. Let T be the largest periodic normal subgroup of G. Clearly everycyclicsubgroup of GrT is almost modular, so that replacing G with ALMOST MODULAR SUBGROUP LATTICES 747

GrT it can be assumed without loss of generalitythat G has no periodic nontrivial normal subgroups. Let x be anyelement of finite order of G, and let X be a subgroup of finite index of G containing x and such that ²:x is a modular subgroup of X. Let N be the core of X in G. Then for each element y of N the finite cyclic subgroup ²:x y is modular in ²:x, N , so that the normal closure ²:x N is periodic byLemma 3.2. It follows that wxN, x is a periodic of G, and hence wxN, x s Ä41. Ž. Therefore N is contained in the centralizer CxG , and the conjugacyclass of x in G is finite, so that ²:x G is finite byDietzmann’s Lemma seeŽ 11,w Part 1, p. 45x. , and hence x s 1. Thus G is torsion-free, and the theorem is proved.

LEMMA 3.4. Let G be a group whose cyclic subgroups are almost modu- lar, and let x and y be elements of infinite order of G such that²: x l ²:y s Ä41.If²: x is a modular subgroup of² x, y :, then xy s yx. Proof. Assume that the statement is false. Since ²:x is modular in ²:x, y , it follows that ²:x is a normal subgroup of ²:Žx, y seew 17, Theorem 1.3x. , and so x y s xy1. On the other hand, ²:y is almost modular in G, and hence there is a positive n such that ²:y is a modular subgroup of ² x n, y:²:². As above y is normal in x n, y:, and xynns yx . This contradiction proves the lemma. The following elementarylemma is probablywell-known, but we have not been able to find it in literature.

LEMMA 3.5. Let G be a residually finite group. Then also the factor group GrZŽ. G is residually finite. Proof. Let ᑥ be the set of all normal subgroups of finite index of G, and let K be anyelement of ᑥ. Then KZŽ. G , G s wxK , G F K , and hence

FFKZŽ. G , G F K s Ä41. Kgᑥ Kgᑥ Therefore F KZŽ. G s ZG Ž., Kgᑥ and so GrZGŽ.is a residuallyfinite group.

LEMMA 3.6. Let G s ²:x, y be a residually finite group with almost modular subgroup lattice. If²: x is a modular subgroup of G, then G is supersoluble. 748 DE GIOVANNI, MUSELLA, AND SYSAK

Proof. ByTheorem 3.3 the set T of all elements of finite order of G is a subgroup. In particular, if x and y have finite order, the group G is periodic and hence even finite byLemma 2.4; as ²:x G is hypercyclically embedded in G, it follows that in this case G is supersoluble. Assume now that one of the elements x and y has finite order and the other has infinite order. As ²:x is a modular subgroup of G, we obtain that either ²:x s T or ²:y s T, so that G is obviouslysupersoluble. Suppose finally that both x and y have infinite order. If ²:x l ²:y s Ä41 , the group G is abelian byLemma 3.4. Thus it can be assumed that ²:x l ²:y / Ä41. Clearly ²:x l ²:y is contained in ZG Ž ., so that the xZ Ž G .and yZŽ. G have finite order; as the factor group GrZG Ž.is residuallyfinite by Lemma 3.5, it follows from the first part of the proof that GrZGŽ.is supersoluble. Therefore the group G is supersoluble.

PROPOSITION 3.7. Let G be a group whose cyclic subgroups are almost modular. If G contains two elements a and b of infinite order such that ²:a l ²:b s Ä41,then G is an FC-group. Proof. Let x be anyelement of infinite order of G, and let X be a subgroup of finite index of G such that ²:x is a modular subgroup of X. Let y be an element of infinite order of X.If²:x l ²:y s Ä41 , we have xy s yx byLemma 3.4. Suppose now that ²:x l ²:y / Ä41 , so that by assumption there exists in X an element z of infinite order such that

²:x l ²:z s ²:y l ²:z s Ä41.

Let Y be a subgroup of finite index of G such that ²:y is a modular subgroup of Y, and let k be a positive integer such that z k belongs to X l Y.As²:y l ²z k : s Ä41 , it follows from Lemma 3.4 that yz kks zy. Then ² y, z k :²:²s y = z k : and yz k is an element of infinite order of X. Since ²:y l ²yz k : s Ä41,we have also ²:x l ²yz k : s Ä41 , and hence

xyz k s yz k x s yxz k .

Therefore xy s yx, and x commutes with all elements of infinite order of X. It follows from Theorem 3.3 that the subgroup X is generated byits Ž. elements of infinite order, so that X is contained in CxG and x belongs to the FC-centre of G. On the other hand, also the group G is generated byits elements of infinite order, and hence it is an FC-group. Proof of Theorem C.Ž. a The set T consisting of all elements of finite order of G is a subgroup byTheorem 3.3, so that replacing G with GrT ALMOST MODULAR SUBGROUP LATTICES 749 we maysuppose that the group G is torsion-free. Let x be anyelement of G, and let X be a subgroup of finite index of G such that ²:x is a modular subgroup of X. Assume that ²:x is not normal in X, and let y be an element of X such that ²:x y / ²:x . Put H s ²x, y :, and let J be the finite residual of H. Since everysubgroup of G is almost modular, the lattice ᑦŽ.J is modular, and so J is an Ž seew 14, Lemma 2.4.9x. . Moreover, the residuallyfinite group HrJ is supersoluble by Lemma 3.6, and hence H is a soluble group. As ²:x is not normal in H, we obtain that ²:x l ²:y / Ä41Ž seewx 16, Theorem 2.²:²: . Thus Hr x l y is finite, so that also the commutator subgroup HЈ of H is finite, and so H is abelian. This contradiction proves that ²:x is a normal subgroup of X. Therefore all cyclic subgroups of G are almost normal, and hence G is an FC-group. As G is torsion-free, it follows that G is abelian. Ž.b Let X be anysubgroup of T, and let K be a subgroup of finite index of G such that X is a modular subgroup of K. Put L s T l K, and let a be anyelement of infinite order of K. Then

X s ²²::²X, a l L s X, a :l L, so that X is normal in ²:X, a .As K is generated byits elements of infinite order, it follows that X is normal in K, and hence it is almost normal in G. Ž.c This part follows directlyfrom Proposition 3.7.

COROLLARY 3.8. Let G be a nonperiodic group with almost modular subgroup lattice. Then GЉ is finite. Proof. Let T be the subgroup consisting of all elements of finite order of G. Then everysubgroup of T is almost normal in G, so that in particular TrZTŽ.is finite Ž seewx 9. and hence TЈ is finite. On the other hand, the factor group GrT is abelian, and so GЉ is finite.

4. PRIMARY GROUPS

Properties of permutable subgroups of infinite groups have been studied byStonehewer 15wx . In particular, he proved that a subgroup H of a group G is permutable if and onlyif H is ascendant in G and it is a modular element of the lattice ᑦŽ.G . It follows that modular subgroups coincide with permutable subgroups in locallynilpotent groups. On the other hand, if G is a periodic group whose subgroups are almost permutable, it is clear that G has no Tarski sections, so that G is locallyfinite byLemma 2.1. Therefore, if p is anyprime number, a p-group G is locallyfinite and has almost modular subgroup lattice if and onlyif all of its subgroups are 750 DE GIOVANNI, MUSELLA, AND SYSAK almost permutable. In this section, we will study p-groups in which every subgroup is almost permutable.

LEMMA 4.1. Let G be a p-group containing an almost permutable sub- group of order p. Then the centre ZŽ. G of G is not tri¨ial. Proof. Let X be an almost permutable subgroup of order p of G, and let K be a subgroup of finite index of G such that X is permutable in K. Then the centre ZKŽ.of K contains a nontrivial element a Žseew 14, Theorem 5.2.9x. , so that a has finitelymanyconjugates in G. It follows that ²:a G is a finite nontrivial normal subgroup of G and thus ZGŽ./ Ä41.

COROLLARY 4.2. Let G be a p-group whose subgroups are almost per- mutable. Then G is hypercentral.

LEMMA 4.3. Let G be a p-group whose centre ZŽ. G has infinite exponent, and let²: x be a permutable subgroup of order p of G. Then x belongs to ZGŽ.. Proof. Clearly x normalizes everysubgroup of G. Let g be any element of G, and let p k be the order of g. Byassumption ZGŽ.contains an element z of order p k. Since x induces a power automorphism on the abelian subgroup ²:g, z , there is a positive integer n such that y xns y for every y g ²:g, z . Thus z nxs z s z, and so also g xns g s g. Therefore x belongs to ZGŽ..

COROLLARY 4.4. Let G be a p-group of infinite exponent whose commuta- tor subgroup GЈ is finite, and let²: x be a permutable subgroup of order p of G. Then x belongs to ZŽ. G . Proof. Since GЈ is finite, the factor group GrZGŽ.has finite exponent. Therefore ZGŽ.has infinite exponent, and x belongs to ZG Ž.byLemma 4.3. The following easylemma suggests that properties of power automor- phisms can be used in the studyof groups with almost modular subgroups.

LEMMA 4.5. Let G be a group, and let X be a modular subgroup of G. If K is a normal subgroup of G such that X l K s Ä41,then e¨ery subgroup of K is normalized by X. Proof. Let y be anyelement of K. Then ²:y s ²y, X :l K is a normal subgroup of ²:y, X , and hence X normalizes all subgroups of K.

LEMMA 4.6. Let G be a periodic residually finite group with almost modular subgroup lattice. Then the commutator subgroup GЈ of G is con- tained in the FC-centre of G. ALMOST MODULAR SUBGROUP LATTICES 751

Proof. Let x be anyelement of GЈ.AsG is locallyfinite byLemma 2.4, there exists a finite subgroup E of G such that x belongs to EЈ. Let K be a normal subgroup of finite index of G such that E l K s Ä41 . Since E is almost modular in G, the subgroup K can be chosen in such a waythat r Ž. E is modular in EK, so that E CKE is isomorphic to a group of power r Ž. automorphisms of K byLemma 4.5. In particular, E CKE is abelian Ž. Ž. and x belongs to CKEG, so that K lies in Cxand the conjugacyclass of x in G is finite. Therefore GЈ is contained in the FC-centre of G. Our next result proves that everyprimaryresiduallyfinite group with almost permutable subgroups has large FC-centre.

PROPOSITION 4.7. Let G be a residually finite p-group whose subgroups are almost permutable, and let F be the FC-centre of G. Then either the fac- r s r , = ¨ tor group G F is cyclic or p 2 and G F C22n C for some positi e integer n. Proof. The commutator subgroup GЈ of G is contained in F by Lemma 4.6, and hence GrF is abelian. Let E be anyfinite subgroup of G, and let K be a normal subgroup of finite index of G such that E l K s Ä41 and E is permutable in EK. Then E induces on K a group of power automorphisms byLemma 4.5. If the subgroup wxE, K is finite, we have obviouslythat E is contained in the FC-centre of EK, and so also in F. Suppose now that wxE, K is infinite. Since everylocallycyclic subgroup of G is cyclic, there exists a finite subgroup V of K such that wxE, V is not cyclic, and the set ᑰ of all such subgroups is a local system ᑰ Ž.s Ž. for K. Moreover, there is an element V0 of such that CKEECV0 , r Ž. Ž² :. and it follows that E CKE can be embedded into Aut y for some Ž wx.Ž. element y of K see 8, Proposition A . Clearly CKE is contained in F, and hence everyfinite subgroup of GrF either is cyclic or isomorphic to = G C22n C for some integer n 1. Assume bycontradiction that the abelian group GrF is infinite, so that it contains a group of type pϱ. Let g be an element of G _ F such that g p g F. Then g p belongs to a finite normal subgroup of G, and so without loss of generalityit can be assumed that g p s 1. Moreover, replacing G with a suitable subgroup of finite index, we maysuppose that ²:g is a permutable subgroup of G. Let N be anynormal subgroup of finite index of G. Then G s GrF l N is a finite-by-abelian group of infinite exponent, and hence g belongs to ZGŽ. byCorollary4.4, so that wxG, g F N.AsG is residuallyfinite, it follows that wxG, g s Ä41 and g belongs to ZGŽ.. This contradiction proves that r = G F is finite, so that it is either cyclic or isomorphic to C22n C for some positive integer n.

LEMMA 4.8. Let G be a p-group whose subgroups are almost permutable. If G s B h A is the semidirect product of an abelian normal subgroup A and 752 DE GIOVANNI, MUSELLA, AND SYSAK

r Ž. a subgroup B, then the factor group G CG A is finite. In particular, if B is abelian, then G is abelian-by-finite. Proof. As the subgroup B is almost permutable in G, there exists a subgroup of finite index A1 of A such that B is a permutable subgroup of ²:s ²:l A11, B . Clearly A A1, B A is normalized by B, and it follows from Lemma 4.5 that B induces on A1 a group of power automorphisms. r Ž. wx Thus B CAB 12is finite bya result of Levi 7 . Let A be a finite sub- s group of A such that A AA12, and let B1be a subgroup of finite index ²:s ²:l of B such that A22is permutable in A , B1; then A2A2, B1A r Ž. is normalized by B11, and so B CAB 2 is finite. It follows that the index < Ž.< 1 Ž.s Ž .l Ž . B : CAB 2 is finite, so that also CABBCA1 CAB 2 has finite r Ž. index in B, and the group G CAG is finite.

LEMMA 4.9. Let G be a residually finite p-group whose subgroups are almost permutable. If G is an FC-group, then GrZŽ. G is finite. Proof. Assume bycontradiction that G is not abelian-by-finite, and suppose that for some positive integer n a subgroup En of G has been defined which is the direct product of n finite non-abelian subgroups. Since G is a residuallyfinite FC-group, there exists a subgroup of finite index K n of G such that l s wxs Ä4 EnnK E nn, K 1.

s Byassumption K nncontains a finite non-abelian subgroup L , and Enq1 = q EnnL is the direct product of n 1 finite non-abelian subgroups. Then the subgroup s E D En ngގ is the direct product of infinitelymanyfinite non-abelian subgroups. By Lemma 2.3.3 ofwx 13 everyfinite nonabelian p-group either has modular subgroup lattice or has a nonabelian section of order p3 which is dihedral for p s 2 and has exponent p for p ) 2. Since in the first case there exists a section isomorphic either to the group Q8 of order 8 or to a semidirect product of two cyclic nontrivial subgroups, and as the direct product of two copies of Q8 has a dihedral section of order 8, it follows that the group G has a section S which is the direct product of infinitely manynonabelian groups each of which can be written as a semidirect product of two abelian subgroups. Thus S itself is the semidirect product of two abelian subgroups, and hence it is abelian-by-finite by Lemma 4.8. This contradiction proves that the FC-group G is abelian-by-finite, and then it is clear that the factor group GrZGŽ.is finite. ALMOST MODULAR SUBGROUP LATTICES 753

COROLLARY 4.10. Let G be a residually finite p-group whose subgroups are almost permutable. Then G is abelian-by-finite. Proof. It follows from Proposition 4.7 and Lemma 4.9. We will now prove that every p-group whose subgroups are almost permutable is a finite extension of an abelian group. To obtain this result, we need information on the behaviour of radicable subgroups of such groups.

LEMMA 4.11. Let G be a locally finite group with almost modular subgroup lattice, and let Q be a subgroup of G ha¨ing no finite nontri¨ial homomorphic images. Then Q is contained in the centre ZŽ. G of G. Proof. Let x be anyelement of G, and put H s ²:Q, x . Suppose first that x has prime order. As ²:x is almost modular in G, we have obviously that ²:x is a modular subgroup of H. Thus either H s Q or Q is a maximal subgroup of H. Moreover, Q is contained in the finite residual of H, so that all subgroups of Q are permutable in H byLemma 2.7. If X is anysubgroup of H which is not contained in Q, then H s QX, and hence X is modular in H. Therefore all subgroups of H are modular, and it follows from the structure of locallyfinite M-groups that H is abelianŽ see wx14, Theorem 2.4.13 and Theorem 2.4.14. . In the general case, let y be an element of prime order of ²:x . Then wxQ, y s Ä41 bythe first part of the proof, and hence y belongs to ZHŽ.. Moreover, byinduction on the order of x we maysuppose that wxQ, x is contained in ²:y , so that wxQ, x is finite r Ž. and it is isomorphic to Q CxQ . Since Q has no finite nontrivial homo- morphic images, it follows that wxQ, x s Ä41 . Therefore Q is contained in ZGŽ..

LEMMA 4.12. Let G be a p-group whose subgroups are almost permutable, and let Q be the largest radicable subgroup of ZŽ. G . Then the factor group GrQ is abelian-by-finite. Proof. Let J be the finite residual of G. Then Q is contained in J, and byLemma 4.11 everynontrivial subgroup of JrQ contains a proper subgroup of finite index. Moreover, the factor group GrJ is abelian-by- finite byCorollary4.10, so that without loss of generalitywe maysuppose that Q s Ä41 and GrJ is abelian. We shall prove that in this situation the group G is abelian. Assume bycontradiction that GЈ / Ä41 , so that in particular G is not residuallyfinite. Then Ž.GЈ p is properlycontained in GЈ because GЈ F J. Replacing G with GrŽ.GЈ p, we mayalso suppose that GЈ has exponent p. Since all subgroups of J Ž.and in particular of GЈ are permutable by Lemma 2.7, GЈ is abelianŽ seewx 14, Theorem 2.4.14. . If GЈ is not contained in ZGŽ., there exists a finite nonabelian subgroup E of G such that 754 DE GIOVANNI, MUSELLA, AND SYSAK wxGЈ, E / Ä41;as E is almost permutable in G and GЈ F J, it follows that E is permutable in GЈE, so that E is even normal in GЈE. Thus the index < Ј Ž.< Ј r Ž.Ј Јr Ј l Ž.Ј G : CEGЈ is finite, and G E ZGE is finite, so that G G ZGE is a finite nontrivial normal subgroup of the hypercentral group GrGЈ l ZGŽ.ЈE . Therefore GЈ contains a G-invariant subgroup N of index p. Clearlythe same propertyalso holds if GЈ lies in ZGŽ., so that replacing G with GrN it can also be assumed that GЈ has order p. Let L be a subgroup of G which is maximal with respect to the condition GЈ l L s Ä41 , so that in particular L is abelian. Since L is almost permutable, replacing G with a suitable subgroup of finite index, it can be assumed that L is permutable in G. Moreover, replacing G with the factor r group G LG , we suppose that L does not contain nontrivial normal subgroups of G, so that L l ZGŽ.s Ä41 . Let h be an element of order p in the set G _ L. Then GЈ l ²:hL/ Ä41 , and so h belongs to GЈL. ⍀ Ž. Ј Therefore the subgroup 1 G is contained in G L, and hence it is ⍀ ŽŽ..s Ј Ž. elementaryabelian. It follows also that 1 ZG G , so that ZG is locallycyclic. Suppose now that G contains two cyclic subgroups ²:x and ²:y such ²:²:/ ²:²: ²:⍀ Ž Ž .. that xy yx. Then x is not normal in G, so that 1 ZG s GЈ is not contained in ²:x and hence ²:x l ZG Ž .s Ä41 . On the other hand, x p belongs to ZGŽ., and so x pps 1. Similarly y s 1, and the subgroup ²:x, y is abelian. This contradiction shows that all subgroups of G are permutable, so that G is an abelian-by-finite group of finite exponentŽ seewx 13, Theorem 2.4.14. . In particular, G must be residually finite, and this last contradiction completes the proof of the lemma.

THEOREM 4.13. Let G be a p-group whose subgroups are almost per- mutable. Then G is abelian-by-finite. Proof. Let Q be the largest radicable subgroup of ZGŽ.. Then the factor group GrQ is abelian-by-finite by Lemma 4.12, and so without loss of generalityit can be assumed that GrQ is abelian. Let M be a subgroup of G which is maximal with respect to the condition Q l M s Ä41 , so that in particular M is abelian. Since M is almost permutable in G, replacing G with a suitable subgroup of finite index we maysuppose that M is a permutable subgroup of G. Suppose that QM is properlycontained in G, and let g be an element of G _ QM such that g p belongs to QM. Clearly Qg²:is abelian, so that Qg ²:s Q = ²:h for some element h of G.As Q l Mh²:/ Ä41 , it follows that Qh²:l M s ²:x is a cyclic nontrivial subgroup, and obviously Qx²:is properlycontained in Qh ²:. Moreover, h p belongs to Qh²:l QM s Qx ²:, so that Qx²:s Qh ² p: and x s zh pn, where z g Q and n is a positive integer which is not divisible by p. Since Q is radicable, there exists an ALMOST MODULAR SUBGROUP LATTICES 755 element y of Q such that z s y p, and so x s Ž.yk pnwhere k s h . Clearly k does not belong to QM, so that yk f QM and Q l Myk²:s ²:u is a cyclic nontrivial subgroup. Therefore Mu²:s Myk ² :, and yk belongs to QM, a contradiction. It follows that G s QM is an abelian group.

COROLLARY 4.14. Let G be a p-group whose subgroups are almost permutable. Then G contains a finite normal subgroup N such that the factor group GrN is quasi-hamiltonian. Proof. The group G is abelian-by-finite by Theorem 4.13, so that G s AE, where A is an abelian normal subgroup and E is a finite subgroup of G. As all subgroups of E are almost permutable in G, A contains a normal subgroup B of G such that the index <

AN l EN s NAŽ.Ž.Ž.l EN s NAl EC s NAl ECs N, and in particular BN l EN s N. Moreover, G s BCE s BEN and so the factor group GrN is the semidirect product of its subgroups BNrN and ENrN. Since N is finite, it is enough to prove that everysubgroup of GrN is permutable, so that without loss of generalityit can be assumed that s h Ž.s Ä4 G E B, where E is core-free and nontrivial. Thus CBE 1 and everysubgroup of B is normal in G byLemma 4.5, so that E acts on B as a group of power automorphisms. Suppose that y is a nontrivial element of E such that b y s by1 for all b g B. Then p s 2 and B contains an element x of order 4, so that the subgroup ²:x, y is dihedral of order 8, a contradiction since ²:y is permutable in ²x, y :. It follows that B has a finite exponent and E is cyclicŽ seewx 1, Theorem 3.5.5. , so that in particular all subgroups of G are permutable if p ) 2Ž seew 14, Theorem 2.4.14x. . Suppose finallythat p s 2, and let U be a cyclic subgroup of order 4 of B. Then the metacyclic group UE is the product of two cyclic permutable subgroups, so that the subgroup lattice ᑦŽ.UE is modular Ž see wx14, Lemma 5.2.14. , and hence EU s E Žseewx 14, Lemma 2.3.6. . Therefore UE s U = E is abelian, and wxU, G s Ä41 , so that all subgroups of G are permutableŽ seewx 14, Lemma 2.3.4. . The corollaryis proved.

5. PERIODIC GROUPS

The first lemma of this section shows that the class of all groups having modular subgroup lattice is local.

LEMMA 5.1. Let G be a group. Then the lattice ᑦŽ.G is modular if and only if e¨ery finitely generated subgroup of G has modular subgroup lattice. 756 DE GIOVANNI, MUSELLA, AND SYSAK

Proof. Let X, Y, Z be subgroups of G such that X F Z, and let x be anyelement of ²:X, Y l Z. Then there exist finitelygenerated subgroups g ²:s ²: X11of X and Y of Y such that x X1, Y1. Since E X 1, Y1has a F s l ²:l modular subgroup lattice and X11Z Z E, we have X111, Y Z s ²:l g ²:l X11, Y Z 1, so that x X, Y Z . It follows that ²:X, Y l Z s ²X, Y l Z :. Therefore the lattice ᑦŽ.G is modular.

LEMMA 5.2. Let G be a locally finite group with almost modular subgroup lattice. Then G contains an abelian normal subgroup A such that the Sylow subgroups of GrA are finite. Proof. Let p be anyprime number, and let S be a Sylow p-subgroup of G. Since S is almost modular in G, there exists a subgroup of finite index X of G such that S is a modular subgroup of X.IfS is permutable in X, l then S is even normal in X, and so S XG is a subnormal subgroup of G which has finite index in S. Suppose now that S is not permutable in X. Ž wx. r Then S is P-embedded in X see 14, Theorem 6.2.17 , and hence S SX r l is finite, so that also S SXGX is finite. In both cases it follows that the r Ž. ᎐ group S OGp is finite. Let R be the Hirsch Plotkin radical of G. Thus the Sylow subgroups of GrR are finite. Since modular subgroups of locally nilpotent groups are permutable, all subgroups of R are almost per- mutable subgroups, and hence for each prime number q the Sylow q-subgroup R q of R contains an abelian normal subgroup of finite index byTheorem 4.13. Then R qqalso has an abelian characteristic subgroup A Ž wx. s of finite index see 6, Lemma 21.1.4 and the direct product A DrqqA is an abelian normal subgroup of G such that all Sylow subgroups of GrA are finite. Ž. LEMMA 5.3. Let G be a group, and let Enng ގ be a sequence of finite subgroups of G with pairwise coprime orders such that all subgroups of Enq1 ²: ¨ ¨ are normalized by E1,...,Enn for each positi e integer n. If e ery E Ž.¨ contains a nonmodular respecti ely, nonpermutable subgroup Hn, then the s ²:N g ގ Ž¨ subgroup H Hn n is not almost modular respecti ely, not almost permutable. in G. Proof. Assume that H is almost modularŽ respectively, almost per- mutable. in G, and let X be a subgroup of finite index of G such that H is a modularŽ. respectively, permutable subgroup of X. Clearlythere exists a positive integer n such that En is contained in X, and byassumption l ²:N / s Ä4 EnkE k n 1. l s Ž. Therefore H EnnH , and H nis a modular respectively, permutable subgroup of En. This contradiction proves the lemma. ALMOST MODULAR SUBGROUP LATTICES 757

We can now prove the following lemma, which will be essential in the proof of Theorem A.

LEMMA 5.4. Let G be a locally finite group with almost modular subgroup lattice. Then G is a finite extension of a group with modular subgroup lattice. Proof. Assume bycontradiction that the locallyfinite group G has no subgroups of finite index with modular subgroup lattice. ByLemma 5.2 G contains an abelian normal subgroup A such that all Sylow subgroups of GrA are finite. Assume first that the factor group GrA is countable. Let n be a positive integer, and suppose that n finite subgroups E1,...,En of G with pairwise coprime orders have been chosen such that everysub- ²:- group of Eiq11is normalized by E ,...,Ei for all i n and the lattices ᑦŽ.ᑦ Ž. s ² : E1 ,..., En are not modular. Since the subgroup E E1,...,En is almost modular in G, there exists a normal subgroup K of G of finite index such that E is a modular subgroup of KE. Let ␲ be the set of all prime numbers dividing the order of the finite subgroup E. As the Sylow subgroups of KrK l A are finite, we have that

r l s Ž.r l K 0 K A OK␲ Ј K A has finite index in KrK l A Žseewx 2, Theorem 3.5.15 and Theorem 2.5.12. , r l and hence replacing K with K 0 it can be assumed that K K A is a ␲ Ј-group. Let P be the ␲-component of the abelian group K l A. As the factor group GrA is countable, there exists a ␲ Ј-subgroup L of K such that K s LP and L l P s Ä41Ž seewx 2, Theorem 2.4.5. . Replacing K with a suitable subgroup of finite index, we mayalso suppose that L is a modular subgroup of K, so that L normalizes all subgroups of P by r Ž. Lemma 4.5, and L CPL is isomorphic to a group of power automor- r Ž. phisms of P. Thus L CPL is finite, so that

Ž.s Ž.= CPKLCP P has finite index in K and so also in G. It follows that the subgroup lattice Ž.ᑦ Ž Ž.. of CPKLis not modular, so that also the lattice CP is not modular, Ž. and byLemma 5.1 there exists a finite subgroup Enq1 of CPL whose ␲ Ј Ž. subgroup lattice is not modular. The -subgroup CPL is characteristic Ž. in CPK , and hence normal in G, so that E normalizes all subgroups of Ž. ␲ Ј CPLnbyLemma 4.5. In particular, the order of E q1 is a -number and ²: all subgroups of Enq11are normalized by E ,...,En . Therefore there Ž. exists a sequence Enng ގ of finite subgroups of G satisfying the state- ment of Lemma 5.3, and hence G contains a subgroup which is not almost modular. This contradiction proves the statement when the group GrA is countable. 758 DE GIOVANNI, MUSELLA, AND SYSAK

We will now prove that the group GrA must be countable. Let XrA be anycountable subgroup of GrA. It follows from the first part of the proof that X contains a subgroup of finite index with modular subgroup lattice, so that in particular X is metabelian-by-finite. On the other hand, the class of soluble-by-finite groups is countably recognizableŽ seew 3, Proposi- tion 2.6x. , so that G itself is soluble-by-finite. Since the Sylow subgroups of GrA are finite, it follows that GrA is countable. The lemma is proved.

LEMMA 5.5. Let G be a locally finite group with almost modular subgroup lattice. If the set of primes ␲ Ž.G is finite, then G contains a finite normal subgroup N such that the subgroup lattice ᑦŽ.GrN is modular. Proof. ByLemma 5.4 the group G is a finite extension of a group with modular subgroup lattice; as the set ␲ Ž.G is finite, it follows that G is abelian-by-finiteŽ seewx 14, Theorem 2.4.13 and Theorem 2.4.14. . As in the proof of Corollary4.14 we can now reduce the proof to the case in which G s E h B, where B is an abelian normal subgroup of G, E is a finite Ž.s Ä4 nontrivial subgroup such that CBE 1 , and H is modular in BH for everysubgroup H of E. In particular, E is a modular subgroup of G. Suppose first that E is permutable in G. Then E G is locallynilpotentŽ see wx14, Theorem 6.3.1. , so that G itself is locallynilpotent. Moreover, it follows from Corollary4.14 that everySylowsubgroup Gp of G contains a ᑦŽ.r finite normal subgroup Nppsuch that the lattice G Npis modular, and s r hence N DrppN is a finite normal subgroup of G and G N has modular subgroup lattice. Suppose now that E is not permutable in G,so that it is P-embedded in G Žseewx 14, Theorem 6.2.17. . Then s = иии = = G S1 St L, where each Si is a non-abelian P-group, elements from different factors have coprime orders, s = иии = = Ž.l E Q1 Qt E L , l each Qiiis a nonnormal Sylow subgroup of S , and E L is a permutable subgroup of G. Clearlythe core of E l L in L is normal in G, so that Ž.l s Ä4 E L L 1 , and it follows from the first part of the proof that the statement holds for the group BEŽ.l L . On the other hand, the set of primes ␲ Ž.Ž.ErE l L and ␲ L are disjoint, so that L is contained in BEŽ.l L and there exists a finite normal subgroup N of L such that ᑦŽ.r = иии = L N is a modular lattice. Since S1 St has modular subgroup lattice, it follows that N is a finite normal subgroup of G such that ᑦŽ.GrN is modular.

LEMMA 5.6. Let G be a locally finite group with almost modular subgroup lattice. Then G s M = K, where the subgroup lattice ᑦŽ.M is modular, the set of primes ␲ Ž.K is finite, and ␲ ŽM .l ␲ Ž.K s л. ALMOST MODULAR SUBGROUP LATTICES 759

Proof. ByLemma 5.4 we have G s AE, where A is a normal subgroup with modular subgroup lattice and E is finite. Moreover, the set of primes ␲ Ž.A can obviouslybe assumed to be infinite, so that

s A Dr An , ngގ where each An is either a nontrivial primarygroup with modular subgroup lattice or a P*-group and elements of different factors have relatively prime ordersŽ seewx 14, Theorem 2.4.13. . For each positive integer n, put

s BnkDr A . kGn

Since E is a finite almost modular subgroup of G, there exists a positive ␲ Ž.l ␲ Ž.s л integer n such that E is modular in EBnnand B E .We wxs Ä4 G claim that Bm, E 1 for some integer m n.If E is permutable in wxs Ä4 BEnn, then E is normal in EB and so Bn, E 1 . Therefore without loss of generalityit can be assumed that E is not permutable in BEn ,so Ž wx. s that E is P-embedded in BEn see 14, Theorem 6.2.17 . Clearly C Ž. CBEnis the core of E in BEn. Hence r s r = иии = r = r BEn C S1 C St C L C, r where each Si C is a nonabelian P-group and elements from different factors have coprime orders, r s r = иии = r = Ž.l r E C Q1 C Qt C E L C, r r l each QiiC is a nonnormal Sylow subgroup of S C, and E L is a wxl s Ä4 s permutable subgroup of BEnn. In particular, B , E L 1 . Put S ²:␲ Ž. G S1,...,St . As the set S is finite, there exists an integer m n such that Bm is contained in L. Then wxl F l s Ä4 Bmm, E S B C 1, and hence wxs Ž.Ž.l l s Ä4 Bmm, E B , E SE L 1. Put

my1 s s M Bmkand K ž/Dr AE. ks1

Then K is normal in G s MK, the set of primes ␲ Ž.K is finite, and ␲ Ž.M l ␲ Ž.K s л. The lemma is proved. 760 DE GIOVANNI, MUSELLA, AND SYSAK

LEMMA 5.7. Let G be a periodic abelian-by-finite group containing a finite normal subgroup N such that the factor group GrN has modular subgroup lattice. Then the lattice ᑦŽ.G is almost modular. Proof. Assume bycontradiction that G contains a subgroup X which is not almost modular, and let A be an abelian normal subgroup of finite index of G. Then X is not almost modular in AX, and so we maysuppose s s Ž.r l that G AX. The centralizer C CAX A X is a normal subgroup of G, and replacing G with the factor group GrC it can also be assumed Ž.s l s Ä4 without loss of generalitythat CAX A X 1 . In particular, the subgroup X is finite, and the counterexample G can be chosen in such a waythat X has minimal order. Write A s U = V, where U is the direct product of the primarycomponents of A of finite exponent and V is that of the primarycomponents of A of infinite exponent. Clearlythe group UX is residuallyfinite, and hence there exists a normal subgroup W of UX contained in U such that UXrW is finite and W l NX s Ä41 . Moreover, the group VXrVX l N has modular subgroup lattice and its Sylow p- subgroups have infinite exponent for each prime p g ␲ Ž.V , so that VXrVX l N is finite-by-abelianŽ seew 14, Theorem 2.4.13 and Theorem 2.4.14x.Ž , and hence the commutator subgroup VX .Ј of VX is finite. It follows that also VXrZVXŽ.is finite, so that V contains a subgroup B of finite index such that wxB, X s Ä41 . Since WB is an X-invariant subgroup Ž.s Ä4 of finite index of A, the minimal choice of X yields that CWBX 1,so that replacing G with WBX we maysuppose that A s W = B, and s s = Ž. Ž.s Ž.s Ä4 l s G AX B WX , with CWXXCA 1 . Clearly WX N X l N centralizes W, so that WX l N s Ä41 and hence the lattice ᑦŽ.WX is modular. In particular, X is a modular subgroup of WX;as X is not modular in G, it follows that the groups B and WX cannot be coprime. Let q be a prime in the set ␲ Ž.B l ␲ ŽWX ., and let Q and Y be the q-components of B and a Sylow q-subgroup of X, respectively. Then Q has infinite exponent, so that also the Sylow q-subgroups of the group QWXrQWX l N have infinite exponent. On the other hand, the lattice ᑦŽ.QWXrQWX l N is modular, so that the Sylow q-subgroups of QWXrQWX l N are central byIwasawa’s theorem, and hence YQWXŽ l N . is a normal subgroup of QWX. Thus

YQWXŽ.l N l WX s YWX Ž.l N s Y

wxs Ä4 Ž. is normal in WX, and so W, Y 1 . Therefore Y is contained in CWX , and this contradiction completes the proof of the lemma. Proof of Theorem A. Suppose first that the lattice ᑦŽ.G is almost modular. It follows from Theorem 2.8 and Lemma 5.6 that G s M = K, where M is a group with modular subgroup lattice and K is a locallyfinite ALMOST MODULAR SUBGROUP LATTICES 761 group such that the set of primes ␲ Ž.K is finite and ␲ ŽM .l ␲ Ž.K is empty. Moreover, Lemma 5.5 yields that K contains a finite normal subgroup N such that the lattice ᑦŽ.KrN is modular. As the group K is a finite extension of a group with modular subgroup lattice byLemma 5.4, we obtain also that K must be abelian-by-finiteŽ seew 14, Theorem 2.4.13 and Theorem 2.4.14x. . Conversely, if G s M = K satisfies the conditions of the statement, it follows directlyfrom Lemma 5.7 that G has almost modular subgroup lattice.

Proof of Corollary D. Assume first that the group G is periodic, so that it is even locallyfinite byTheorem 2.8. As locallyfinite groups with modular subgroup lattice are metabelian, it follows now from Theorem A that G is metabelian-by-finite. Suppose now that G is not periodic, and let T be the subgroup consisting of all elements of finite order of G. Then all subgroups of T are almost normal in G byTheorem C, and in particular TrZTŽ.is finite. If G is an FC-group, there exists a finite normal s Ž. Ž.s l Ž . subgroup N of G such that T ZT N, so that ZT T CNG ;it follows that GrZTŽ.is abelian-by-finite, and hence G is metabelian-by- finite. Assume now that G is not an FC-group, so that the factor group r s ŽŽ..r G T is locallycyclic.The centralizer C CTG ZT is a normal subgroup of G, and GrC is finite. Moreover, C l TrZTŽ.is contained in the centre of CrZTŽ., and CrC l T is locallycyclic,so that CrZT Ž.is abelian and G is metabelian-by-finite.

ReplacingŽ. almost modular subgroups withŽ. almost permutable sub- groups, the proof of Lemma 5.7 can be adapted to prove the following result.

LEMMA 5.8. Let G be a periodic abelian-by-finite group containing a finite normal subgroup N such that the factor group GrN is quasi-hamiltonian. Then all subgroups of G are almost permutable.

Proof of Theorem B. Suppose first that all subgroups of G are almost permutable, so that G is locallyfinite. ByTheorem A we have that G s M = L, where M is a group with modular subgroup lattice, L is an abelian-by-finite group containing a finite normal subgroup W such that the lattice ᑦŽ.LrW is modular and ␲ Ž.Ž.M l ␲ L s л. Therefore M s Q = E and LrW s UrW = VrW, where Q and UrW are locallynil- potent groups, E and VrW are direct products of P*-groups, and ␲ Ž.Q l ␲ Ž.E s л Žseewx 14, Theorem 2.4.13. . Since every P*-group contains a subgroup which is not permutable, it follows from Lemma 5.3 that E and VrW are finite. Clearly Q is a quasi-hamiltonian group, and N s EV is a finite normal subgroup of the abelian-by-finite group 762 DE GIOVANNI, MUSELLA, AND SYSAK

K s E = L such that KrN , UrW is quasi-hamiltonian. Moreover, G s Q = K and ␲ Ž.Q l ␲ Ž.K s л. Conversely, if G s Q = K satisfies the conditions of the statement, application of Lemma 5.8 yields that all subgroups of G are almost permutable. It follows from Theorem A that everyperiodic group G with almost modular subgroup lattice contains a finite normal subgroup N and a subgroup of finite index M such that the lattices ᑦŽ.GrN and ᑦ Ž.M are modular. The consideration of the direct product G s E = T, where E is a group of prime order p and T is a Tarski p-group, shows that the above condition is not enough to prove that G has almost modular subgroup lattice. On the other hand, a characterization of this type can be given for periodic groups whose subgroups are almost permutable.

COROLLARY 5.9. Let G be a periodic group. Then all subgroups of G are almost permutable if and only if G is finite-by-quasi-hamiltonian and quasi- hamiltonian-by-finite. Proof. The necessityof the condition is an obvious consequence of Theorem B. Suppose converselythat the periodic group G contains a finite normal subgroup N and a normal subgroup L of finite index such that GrN and L are quasi-hamiltonian groups. Consider the of primes ␲ s ␲ Ž.N j ␲ ŽGrL ..AsGrN is locallynilpotent, we have

GrN s HrN = KrN, where HrN is a ␲ Ј-group and KrN is a ␲-group. Then H s QN for some ␲ Ј-subgroup Q, which is of course quasi-hamiltonian. Clearly G s HK s QK and ␲ Ž.Q l ␲ Ž.K s л;asQ is contained in the locally L, it follows that

L s QKŽ.l L s Q = Ž.K l L , so that Q is a normal subgroup of G and G s Q = K. Moreover, the subgroup K is abelian-by-finite, since K l L is quasi-hamiltonian and the set ␲ Ž.K is finite. Therefore all subgroups of G are almost permutable by Theorem B.

LEMMA 5.10. Let G be a locally finite group with almost modular subgroup lattice. If the set of primes ␲ Ž.G is finite, then G contains an abelian normal subgroup of finite index A such that all subgroups of A are normal in G. Proof. ByLemma 5.4 the group G contains a subgroup of finite index with a modular subgroup lattice, so that G is abelian-by-finiteŽ seew 14, ALMOST MODULAR SUBGROUP LATTICES 763

Theorem 2.4.13 and Theorem 2.4.14x. and hence G s VE, where V is an abelian normal subgroup of G and E is a finite subgroup. As E is almost modular in G, there exists a subgroup of finite index W of V such that W is normal in G and E is a modular subgroup of WE. Suppose first that E s = is a p-group for some prime number p and put W W12W , where W 1 l s Ä4 is the p-component of the abelian group W. Then W2 E 1 , and it follows from Lemma 4.5 that all subgroups of W2 are normalized by E and so are normal in G. ByLemma 5.5 there exists a finite normal subgroup N ᑦŽ.r r of WE111such that the lattice WE N is modular. If WE N is abelian, Ž.Ј Ž. we obtain that WE1 is finite, and hence the centralizer CEW has finite s Ž. 1 index in W1; in this case it is enough to put A CEWW 2 . Assume now r 1 Ž that the p-group WE1 N is not abelian, so that it has finite exponent see wx. 14, Theorem 2.4.14 . Then WE1 is residuallyfinite, and there exists a r l s Ä4 normal subgroup U of WE11such that WE U is finite and U E 1; again byLemma 4.5 everysubgroup of U is normalized by E, so that the s abelian subgroup A UW2 has the required properties. Suppose finally that the order of E is not a power of a prime, and for each prime g ␲ Ž. p E let Ep be a Sylow p-subgroup of E. It follows from the first part of the proof that W contains a subgroup of finite index Appsuch that E normalizes all subgroups of Ap. Then

s A F Ap pg␲ Ž.E is an abelian normal subgroup of finite index of G all of whose subgroups are normal in G. The lemma is proved.

THEOREM 5.11. Let G be a periodic group with almost modular subgroup lattice. Then G contains a normal subgroup of finite index M such that all subgroups of M are modular in G. Proof. It follows from Theorem 2.8 and Lemma 5.6 that G s H = K, where H is a group with modular subgroup lattice, K is a locallyfinite group with ␲ Ž.K finite, and ␲ Ž.H l ␲ Ž.K s л. Moreover, Lemma 5.10 yields that K contains an abelian normal subgroup A such that KrA is finite and all subgroups of A are normal in K, and so also in G. Thus the normal subgroup M s HA of G satisfies the condition of the statement.

COROLLARY 5.12. Let G be a periodic group whose subgroups are almost permutable. Then G contains a normal subgroup of finite index Q such that all subgroups of Q are permutable in G. Proof. ByTheorem 5.11 there exists a normal subgroup of finite index M of G such that all subgroups of M are modular in G. Moreover, it follows from Corollary5.9 that M contains a normal subgroup Q such that 764 DE GIOVANNI, MUSELLA, AND SYSAK

Q is quasi-hamiltonian and MrQ is finite, and clearly Q can be chosen normal in G. Then everysubgroup of Q is ascendant and modular in G, and so also is permutable.

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