This paper deals with one of the ways of studying infinite groups many of whose subgroups have a prescribed property, namely the consideration of minimal conditions. If P is a theoretical property of groups and subgroups, we show that a locally graded group G satisfies the minimal condition for subgroups not having P if and only if either G is a Cernikov group or every subgroup of G satisfies P, for certain values of P concerning to normalilty, nilpotency and related ideas

Otal, Javier

Peña, Juan Manuel

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Publicacions Matemátiques, Vol 32 (1988), 151-157.AbstractLOCALLY GRADED GROUPS WITH CERTAINMINIMAL CONDITIONS FOR SUBGROUPS, IIJAVIER OTAL AND JUAN MANUEL PEÑAThis paper deals with one of the ways of studying infinite groups manyof whose subgroups have a prescribed property, namely the considera-tion of minimal conditions . If P is a theoretical property of groups andsubgroups, we show that a locally graded group G satisfies the minimalcondition for subgroups not having P if and only if either G is a Cer-nikov group or every subgroup of G satisfies P, for certain values of Pconcerning to normalilty, nilpotency and related ideas.1 . IntroductionThis paper deals with one of the most important topics in the study of infinitegroups many of whose subgroups have a prescribed property, namely the con-sideration of minimal conditions in the direction described by the introductionof [9] . Specifically we shall prove :Theorem . Let c be a nonnegative integer and let P be one of the followingproperties : (i) normal; (ii) locally nilpotent; (iii) nilpotent of class at most c;(iv) normal or locally nilpotent; (v) normal or nilpotent of classc-hamiltonian. If G is a locally graded group, then we have :(1) A minimal non-P subgroup M of G is finite .(2) G satisfies the minimal condition for subgroups not having P if andonly if either G is a Cernikov group or every subgroup of G satisfies P.al most c; (vi)A group G is said to be locally graded if every non-trivial finitely generatedsubgroup of G has a non-trivial finite image . This weak finiteness conditionis made in order to avoid the presence of the Tarski's monsters or the groupsdescribed by O1'sanskii in [5], which show that the result is false without somesort of finiteness condition . It is easy to show that the class of locally gradedgroups is a local and a residual class of groups and, since abelian and finitegroups are clearly locally graded, it follows that such class is very extensive asit contains many different types of groups (see[2, p . 155]) .152	J . OTAL, J.M. PEÑAThe second part of the present Theorem is an extension of some cases of[9, Theorem B], where the same conclusion in cases (i)-(v) was proved for W---groups, that is, groups in which every finitely generated non-nilpotent subgrouphas a finite non-nilpotent image. This caass of groups is subgroup and locallyclosed, closed under extensions and it is also very ample because it includeslocally hyper-(abelian or finite) groups and linear groups . However W-groupsform a proper subclass of locally graded groups ; indeed the groups constructedin [11] are locally graded but they are not W-groups . In [6] and especially in[7] we have initiated the aforesaid extension from W-groups to locally gradedgroups adding the e-hamiltonian case, which has a different behaviour and willbe introduced and explained in the next section . Thus the main result in [7]shows completely the case (i) and the cases (iii), (v) and (vi) when c = 1 .To do this, a fundamental tool is the finiteness of the minimals, which wehave included as our first conclusion because it is of interest in its own right .Precedents of that were some of the results obtained in [1] (Corollaries 1 and2), where the finiteness of a minimal in case (iii) for locally graded groups andin case (ii) for W-groups was proved ; the extension of the latter to locallygraded groups is immediate since a such minimal is finitely generated and sonilpotent-by-finite . Finally, Ivan [4] has studied the locally nilpotent and theabelian cases under various assumptions, some of which are included in ours .Throughout our general group-theoretical notation is standard and is takenfrom [10] . JV, will denote the caass of nilpotent groups of caass at most e ; thusG E N, if and only if y,+i(G) = 1 . The definitions of minimal non-T> groupor group satisfying the minimal condition for subgroups not having P (in shortMin(non-P)) can also be interpreted within that context (see [1], [2], [6], [7],[8] or [9]) . For our convenience we quote here some well-known facts aboutnilpotent groups, which will be freely used in what follows : Le¡ G be a nilpotentgroup. Then G is finite if and only if GIG is finite ([10] ; Corollary to 2 .26) . Ifmoreover G is finitely generated, then G is polycyclic and so it is a residuallyfinite group satisfying Max and contains a characieristic torsion-free subgroupof finite index ([10] ;Part 1 pp . 65-66, Corollary to 9.31 and Part 2 pp . 138-139) . Iffurthermore G is torsion-free, then G is residually a finite p-group forevery prime p ([10, Corollary to 9.38]) .2 . c-Hamiltonian groupsIn a certain sense, this type of groups was first considered in [1], at leastwith this degree of generality. Let us go to the definitions . If c is a nonnegativeinteger, a group G is said to be a c-hamiltonian group if every subgroup of Gis either normal in G or an N,-group . Thus O-hamiltonian groups are exactlyhamiltonian (or Dedekind) groups and their structure is well-known by resultsdue to Dedekind and Baer (see[1, 2.1.2],for example), and 1-hamiltonian groupsare usually known as metahamiltonian groups . Cleaxly the c- hamiltoniangroups form a caass which is closed under subgroups and quotients . It is notLOCALLY GRADED GROUPS	153hard to show that this class is also locally closed, which will be fundamentalin what follows . For, assume that H is locally a c-hamiltonian group thatcontains a subgroup K which is not normal in H and such that -y,+l (K) 7~ 1 .Then K contains a finitely generated non-,V, subgroup, say L, and there existelements x in K and h in H such that xh 1 K. The subgroup (L, h, x) ofHis c-hamiltonian so that its subgroup S = (L, x) must be normal in it . In particularxh E S, which is a contradiction showing our claim . A similar argument givessome information about the form of a minimal c-hamiltonian group G: it canbe generated by a finitely generated non-N, subgroup, say L, and by elementsx and h in G such that x h 1 (L, x) .Only a few facts are known about c-hamiltonian groups in general . Forexample a locally graded c-hamiltonian group is a W-group ([1, Proposition2]) . Since there are locally graded 1-hamiltonian groups which are not locallynilpotent ([1, Theorem D and VII]), we cannot include c-hamiltonian groupsamong the classes considered by Phillips-Wilson (see [9, p . 45]) and we shallneed to provide dif£erent proofs to solve this case . This will be carried out withthe aid of later classification results as those of [2] and [8] . To apply thereresults we have needed to show a property of c-hamiltonian groups, which isinteresting in its own right . The case c = 1 was originally showed by Cernikovin [3, p . 511] and later by Bruno-Phillips in [1, Corollary 3] .(2.1) . Leí H be a locally graded c-hamiltonian group. Then either H belongsto N, or H' is finite .Proof. We assume that H 1 N, and then we have to show that H' is finite .Since H is a W-group, [1, Theorem B] assures that H' is finite in the casewhere H is not locally nilpotent . Thus we assume that H is locally nilpotent .First at all, we claim that if H is finite, then jH'j is divisible by at mosttwo different primes (one of which is necessarily 2) . For, if N is a minimalnon-N, subgroup of H, then N is a normal p-subgroup of H and HIN ishamiltonian . Therefore the derived group of HIN has order at most two, thatis, IH'NINI < 2 and our claim follows.Assume that H is infinite and let K be a finitely generated non-N, subgroupof H. Then again jH'KIKI < 2 . Furthermore K is nilpotent and so H'nK andH' are finitely generated . Thus we may choose a finitely generated subgroup Vof H containing K such that V' = H'. Therefore, replacing H by V, we mayassume that H is finitely generated and so nilpotent . Since HIH' is infinite,H contains a torsion-free normal subgroup L such that H/L is finite . For eachinteger m >_ 1, H/L' is a finite nilpotent group and, since L is residually finiteand H q X. we see that some HIL' 1N, Thus, replacing L by this L', wemay assume that HIL V A/, . As we remarked above, J(HIL)'j is divisible by atmost two primes . It is not hard to see that, replacing L by some adequate powerif necessary, we may choose a prime q such that j(H/L9n y1 is a q'-number forevery n > 0 . Since L/L9n is always a . q-group, it follows that H' nL = H' nL9n154	J. OTAL, J .M . PEÑAfor every n >_ 0 and then H' n L = 1 because L is residually a finite q-group .Therefore H' is finite .3. Proof of the TheoremProof of (1) : As we said in the introduction, it remains to show the cases(iv)-(vi) . In any case we note that M is finitely generated . For our convenientewe fix some additional notation : In what follows ifp is a prime, Mp will denote anormal subgroup of M of index p. The existente of images of M with arbitraryfinite orders will be assured in case of M/M' being infinite, because, in such acase, M/M' will have an infinite cyclic image .Cases (iv) ajad (v) . Assume that M is infinite . Choose x in M and g in Gsuch that x9 1 M. By the case (ii) M contains proper non locally nilpotentsubgroups and if H is one of them, then H is normal in G, M = H(x~ byminimality and so M' <_ H. Clearly MIM' must be infinite ; otherwise a nonlocally nilpotent subgroup K of M with M' <_ K < M and [M : K] maximalwould be finite by the case (ii) . Thus M has finite images with arbitrary orders .If p and q are different primes, then M = Mp M, so that at most one of theseMP can be normal in G and therefore the others are locally nilpotent (in factnilpotent, in case (iv), and N,-groups, in case (v)) . Hence M is nilpotent . Incase (iv), that is a contradiction . In case (v), let MIL be a finite non-N, imageof M; the existente of a such image is ensured because M is residually finite .Pick a prime q not dividing [M : L] and such that Mq E JV, . Then M = Lllilqand so MIL E .V,, a contradiction .Case (vi) . By (2.1) every proper subgroup of M is either an JV,-group or afinite-by-abelian group . Since M is finitely generated,[2, Theorem 2] or [8, 1 .4and Theorem 1] give that M is a finite-by-Nd group where d = c, if c >_ 1, andd = 1, if c = 0 . Thus yd+l (M) is finite and M17d+1(M) is a finitely generatednilpotent group . It follows that M is polycyclic-by- finite and, in particular,M is a residually finite group satisfying Max .Assume that M is infinite . Then MIyd+l (M) is infinite and it follows thatMIM' is infinite . Again M has finite images with arbitrary orders . SinceM 1 N,, it has a finite non-N, image, say M/K. If p is a prime not dividing[M : K], then M = KMp and then Mp cannot be an N,-group . By (2.1)(Mp )' is finite, Mp is central-by-finite ([10, 4.32 and 4.35]) and MICM(Mp )is finite . Given p and q different primes not dividing [M : K], we have thatM = Mp M, and, since MICm(Mp ) and MICM(Mq) are finite, we find thatM/«M) is finite. Let MIL be a finite image of M whose order is coprime toIM : «M) 1 ; then M = L«M). Since M is not e-hamiltonian, we may choosea subgroup U of M which is neither normal in M nor an JV,-group . By theco-primeness condition, U C_ (U n L)(U n «M)) mod L n «M) and from thisit follows that U = (U n L)(U n «D7)). Thus U n L is neither normal in Mnor an ,N,-group . This is a contradiction because L is c-hamiltonian so thatU n L must be normal in L and this gives that U n L is normal in M.LOCALLY GRADED GROUPS	155Thus the proof of (1) is now complete . The proof of (2) needs some auxiliaryresults which we now state . Some of them are, no doubt, known : we includestatements here for the reader's convenience without proof .The first fact is a general property of locally graded groups, which can befound in [6, Lemma 1] in an equivalent form .(3.1) . A finitely generated infinite locally graded group has an infinite des-cending chain of normal subgroups offinite índex.As a consequence we have that a finitely generated locally graded group G isfinite if and only if G/F(G) is finite where F(G) is the finite residual of G,that is, the intersection of all normal subgroups of G of finite index .The next results is due to Cernikov and a proof of it can be found in [9,Lemma 20] .(3.2) . If M is a finite subgroup of a group G and (Gil¡ E I) is a descendingchain of M-invariant subgroups of G, thenn(GiMii E I) = (n(Gi li E I))M .The last auxiliary result allows us to easily reduce some cases from locallygraded groups to W-groups .(3.3) . LeíX be a subgroup closed caass of groups and let G be a locally gradedgroup satisfying Min (non-LX) that contains a (non-trivial) minimal non-LXsubgroup M, which is finite . Then G is a locally finite group satisfying Min-n .Proof. Assume that G has an infinite descending chain (Gn in E N) of normalsubgroups . Since M is finite, it is easy to show that the descending chain(GnMIn E N) is strict unless at only finitely many steps . Therefore, for somen, GnM is an LX-group and so is M, a contradiction which shows that Gsatisfies Min-n .Now let H be a finitely generated subgroup of G. As above we find that(H,M) satisfies Min-n so that it must be finite by (3.1) . Therefore H is finiteand then G is locally finite .Proof of (2) : The cases (ii) and (iii) can be reduced to W-groups using thepart (1) and (3.3) so that they follow from [9, Theorem B].Cases (iv) and (v). Put X = locally nilpotent, in case (iv), and X = Nc,in case (v) . Assume that G has a minimal non normal non- LX subgroupM, which is finite by (1) . We claim that G is a W-group so that our resultwill follow from [9, Theorem B] again . Assume that G is not a W--group andtake H to be a finitely generated non-W subgroup of G ; we may replace H by(H, M) to assume that M <_ H. Evidently, H is infinite : from (3.1) H containsan infinite descending chain (H,, in E N) of normal subgroups of finite index .156	J . OTAL, J .M . PEÑASince H is not a W-group and X is a caass of W-groups, no term of the abovechain can be an X-group so that we may assume that every H,, is normal in G .Let (Gi Ii E I) be the set of all non-trivial normal subgroups of G and let N bethe intersection of all them . The cardinal of the family (GiMii E I) is infiniteand the intersection of its members is NM by (3.2) . Obviously no GiM canbe an LX-group so that we may assume that every GiM is normal in G andtherefore NM is normal in G and we note that NM is contained in H.Clearly N satisfies Min (non-LX) . If N is not an LX-group, then N islocally finite by (3.3) and (1) . In any case N is a W-group and so is HM.Now H is not a W-group so that H/NM cannot be a W-group . Since H/NMsatisfies Min (non-normal), [7, Lemma 3] gives that H/NM is periodic .Let F be the finite residual of H . As above, using (3.2) within H, we findthat FM is normal in G . In particular N <_ FM and so HIFM is periodic,HIFM is isomorphic to the quotient of a residually finite group by a finitegroup and therefore H/FM is residually finite, which can be shown by using(3.2) . In particular H/FM is locally graded . Since HlFM also satisfies Min(non-normal), [7, Theorem (ü)] gives that HlFM is either abelian or locallyfinite and therefore, in any case, H/FM is finite . Hence HIF is finite and sois H by the consequence quoted after (3.1), a contradiction which shows ourclaim and proves these cases .Case (vi) . By (2.1) G satisfies the minimal condition for non finite-by-.Ndsubgroups where d = c, if c >_ 1, and d = 1, if c = 0 . By [2, Theorem 1] or [8,Theorem 1], yd+i(G) is Cernikov . If G is not c-hamiltonian, then G satisfiesMin-n by (3.3) and so every locally nilpotent quotient of G is Cernikov (see[10, Corollary 2 to 5.27]) . In particular G1yd+l(G) is Cernikov and so is G .Referentes1 . B . BRUNO, R . E . PHILLIPS, Groups with restricted non-normal subgroups,Math . Z . 176 (1981), 199-221 .2 . B . BRUNO, R . E . PHILLIPS, On minimal conditions related to Miller-Mo-reno type groups, Rend . Sem . Mat . Univ.Padova 69 (1983), 153-168 .3 . S . N . C ERNIKOV, Infinite nonabelian groups in which all infinite nonabeliansubgroups are invariant, Ukrain . Math . J . 23 (1971), 498-517 .4 . L . IVÁN, On groups satisfying the minimal condition for certain subgroups,Period . Math . Hungar 14 (1983), 117-124 .5 . A . YU . OL'SANSKII, Infinite groups with cyclic subgroups, Soviet Math .Dokl 20 (1979), 343-346 .6 . J . OTAL, J . M . PEÑA, Some minimal conditions for locally gradedgroups, Pub . Sem . Mat . G . Galdeano serie II, 71 (1985) .7 . J . OTAL, J . M . PEÑA, Grupos localmente graduados con ciertas condicio-nes minimales sobre subgrupos, Actas VII Congresso GMEL Coimbra ILOCALLY GRADED GROUPS	157(1985),251-254 .8 . . J . 0TAL, J . M . PEÑA, Groups in which every proper subgroup is Cernikov-by-nilpotent or nilpotent-by-Cernikov, Arch . Math., to appear .9 . R .E . PHILLIPS, J . WILSON, On certain minimal conditions for infinitegroups, J. Algebra 51 (1978), 41-68 .10 . D .J .S . ROBINSON, Finiteness conditions and generalized solublegroups, Springer-Verlag (1972), Berlin .11 . J .S . WILSON, On periodic generalized nilpotent groups, Bull . LondonMath . Soc . 9 (1977), 81-85 .J . Otal : Departamento de MatemáticasFacultad de CienciasUniversidad de Zaragoza50009 Zaragoza, SPAIN .J.M . Peña : Departamento de Matemática AplicadaEscuela U de ITIUniversidad de Zaragoza50009 Zaragoza, SPAIN .Rebut el 8 d'Octubre de 1987

Locally graded groups with certain minimal conditions for subgroups, II

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Locally graded groups with certain minimal conditions for subgroups, II

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