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Groups with Almost Modular Subgroup Lattice Provided by Elsevier - Publisher Connector Journal of Algebra 243, 738᎐764Ž. 2001 doi:10.1006rjabr.2001.8886, available online at http:rrwww.idealibrary.com on View metadata, citation and similar papers at core.ac.uk brought to you by CORE Groups with Almost Modular Subgroup Lattice provided by Elsevier - Publisher Connector 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 Mathematics, 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 group G is called modular if it is a modular element of the lattice ᑦŽ.G of all subgroups 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 order. 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 isomorphism from the lattice ᑦŽ.G onto the subgroup lattice of a group G and N is a normal subgroup of G, then the image 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-finite group 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 lattice of subgroups 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 distributive lattice 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.
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