PRE-PUBLICACIONES del seminario matematico 2002 Groups with a few non-subnormal subgroups L.A. Kurdachenko J. Otal garcia de galdeano n. 16 seminario matemático garcía de galdeano Universidad de Zaragoza
Groups with few non subnormal subgroups LEONID A. KURDACHENKO Department of Algebra, University of Dnepropetrovsk Vul. Naukova 13. Dnepropetrovsk 50, UKRAINE 49050 mmf@ff.dsu.dp.ua JAVIER OTAL Departamento de Matemáticas, Universidad de Zaragoza 50009 Zaragoza, SPAIN otal@posta.unizar.es Abstract In this paper we review the structure of groups that satisfy the minimal or maximal conditions on their normal or subnormal subgroups. 2000 Mathematics Subject Classification: 20E15. To Alfredo R-Grandjean in his passing away 1 Introduction Let ν be a theoretical property of groups or subgroups. This means that we allow ν to be either a class X of groups, as abelian groups, nilpotent groups and so on, or a theoretical property of a subgroup of a group as being a normal subgroup, a subnormal subgroup, an almost normal subgroup or a permutable subgroup for example. Given a group G, let L non ν (G) = {H G H has no the property ν}. This research was supported by Proyecto BFM2001-2452 of CICYT (Spain) and Proyecto 100/2001 of Gobierno de Aragón (Spain) The contents of this paper corresponds to a lecture given in May 2002 in the University of Zaragoza when the first author was a Visitor Professor of the same 1
One of the first important problems in Theory of Groups is the study of the influence of the family L non ν (G) on the structure of the group G itself for the most important natural properties ν. The starting point of such researches lies in the paper of Dedekind [7], where finite groups with all subgroups normal are classified, that is the family L non normal (G) is supposed to be empty. The general structure of such groups was described by Baer [1]. The next important paper in this direction was the paper of Miller and Moreno [22], where finite groups in which every proper subgroup is abelian were considered. This is the case L non ab (G) = {G}. In the paper of O. Yu. Schmidt [26], finite groups with all proper subgroups nilpotent ( that is L non nil (G) = {G}) have been described. Further, Schmidt began to study the groups G in which the set L non ν (G) is very small in some sense. In particular, in his paper [27], he described those finite groups G in which the subgroups of the family L non norm (G) are all conjugate; more, in the paper [28] he carried out the description of the finite groups in which the family L non norm (G) is the union of exactly two conjugacy classes. In the above examples, G is a finite group and the set L non ν (G) clearly has few elements. But, what means that L non ν (G) has few elements or L non ν (G) is very small when G is an infinite group? Among the different possibilities of giving sense to this, S. N.Chernikov proposed the following approach. Consider those infinite groups G for which a family of the form L non ν (G) satisfies some finiteness conditions, in particular the maximal or the minimal condition on the members of the family. This approach has appeared very interesting and successful. Many papers have been written concerning properties ν in which the set L non ν (G) satisfies the most natural finiteness conditions for certain choices of ν. The goal of this survey paper is to review the most important results obtained for the families L non norm (G) of all non normal subgroups of G and L non sn (G) of all non subnormal subgroups of G. 2 Groups with a small set of non normal subgroups We have already remarked that the finite groups all subgroups of which are normal were described by Dedekind [7]. His results were extended to arbitrary groups by Baer [1]. Such groups are now known as Dedekind groups. They have a nice and simple structure: either a Dedekind group G 2
is abelian or G = A B Q, where A is a periodic abelian 2 group, B is an elementary abelian 2 group and Q is a quaternion group. Related to this, S. N. Chernikov [4] has considered those groups G whose infinite subgroups are all normal. In other words, this the case in which the family L non norm (G) of all non normal subgroups of G consists of finite groups only. Quoting results of S. N. Cernikov [3] and Shunkov [33], we may collect the features of these groups. Theorem 2.1 Let G be an infinite group, all infinite subgroups of which are normal. (1) If G is non-abelian, then G is periodic. (2) If G is locally finite, then either G is a Dedekind group or G includes a normal Prüfer subgroup K such that G/K is a finite Dedekind group. We mention that the condition of local finiteness cannot be removed in this theorem. Indeed there exist infinite periodic groups with all proper subgroups finite. Examples of such groups were constructed by A. Yu. Ol shanskij in [24, chapter 28]. Furthermore, S. N. Chernikov [4] considered the groups G, in which the family L non norm (G) satisfies the minimal condition, that is the groups with the condition Min (non-norm). Theorem 2.2 Let G be an infinite group satisfying Min (non-norm). (1) If G is not periodic, then G is abelian. (2) If G is locally finite, then either G is a Dedekind group or G is a Chernikov group. Dual to the minimal condition is the maximal condition. The groups in which the family L non norm (G) satisfies the maximal condition ( that is, the groups with the condition Max (non-norm)) were considered by Kurdachenko, Kuzenny and Semko in [?] and Cutolo [5]. The basic results of the above papers can be stated in the following way. Theorem 2.3 Let G be a locally graded group satisfying Max (non-norm). Then G has one of the following types: (1) G is a polycyclic-by-finite group, or 3
(2) G is a Dedekind group, or (3) the center ζ(g) of G includes a Prüfer p subgroup P such that G/P is a finitely generated Dedekind group, or (4) G = H L, where H Q 2 (the additive group of the dyadic numbers) and L is a finite non abelian Dedekind group. If every finitely generated subgroup of a group G is normal, then every subgroup of G is also normal. Therefore it is interesting to look at the dual situation. What we can say about the structure of a group G, every not finitely generated subgroup of which is normal. This is the case in which the family L non norm (G) consists of finitely generated subgroups only. Such groups have been considered by Kurdachenko and Pylaev [15], Cutolo [5], Cutolo and Kurdachenko [6]. The basic results of these papers can be stated in the following way. Theorem 2.4 Let the group G have an ascending series, every factor of which is locally (soluble-by-finite group). Then every not finitely generated subgroup of G is normal if and only if G has one of the following types: (1) G is a Dedekind group, or (2) G includes a Prüfer subgroup P such that G/P is a finitely generated Dedekind group, or (3) G satisfies the following conditions: (3a) the center ζ(g) of G includes a Prüfer p subgroup K such that G/K is a minimax abelian group with finite periodic part, (3b) the spectrum Sp(G/K) = {p}, (3c) G/F C(G) is a torsion-free group, and (3d) if A is an abelian subgroup of G, then A/(A K) is finitely generated. (4) G = T A, where A = Q 2 and T is a finite Dedekind group, or (5) G satisfies the following conditions: (5a) G = (A T ) < g >, where A = Q p (the additive group of the p adic numbers) for some prime p and T is a finite Dedekind subgroup, (5b) if T is non-abelian, then p = 2, (5c) the element g induces a power automorphism on the Sylow p subgroup T p of the subgroup T, and (5d) there is a number r 1 such that a g = a c, where c = ±p r for each a AT p, where T p p is the Sylow p subgroup of T. On the other hand, Baer [2] and Zaitsev [34] have introduced new interesting finiteness conditions, namely the weak maximal and the weak minimal 4
conditions for different families of subgroups. The results of Baer and Zaitsev have generated a large series of papers, in which groups with weak minimal and maximal for different types of subgroups have been considered. Here we focus in the results connected with the family of non-normal or non-subnormal subgroups. Let M be a given family of subgroups of a group G. We say that M satisfies the weak minimal condition or the group G satisfies the weak minimal condition for M subgroups (in short, G satisfies the condition Min M) if G has no infinite descending series {H n n 1} such that the indexes H n : H n+1 are infinite for every n 1. Replacing descending series by ascending series, we obtain the dual concept, that is G satisfies the condition Max M). In particular, if M = L non norm (G),the group G is said to satisfy Min (non-norm) and Max (non-norm), respectively. These groups have been studied by Kurdachenko and Gorezky [14]. Their main result is the following Theorem 2.5 Suppose that G is a locally (soluble by finite) group.then G satisfies the condition Min (non-norm) (respectively, Max (nonnorm)) if and only if either G is a Dedekind group or G is a minimax group. 3 Groups with a small set of non subnormal subgroups If G is a finite group, it is well known that if every subgroup of G is subnormal (that is L non sn = ), then G is nilpotent. This result is far from be true when G is infinite. For example, there exist infinite locally nilpotent groups with identity center, all subgroups of which are subnormal. Examples of these groups have been constructed by Heineken and Mohamed [11, 13], Hartley [9] and Menegazzo [21]. The general behaviour of the groups with all subgroups subnormal has been studied by many authors and is collected in the excellent book of Lennox and Stonehewer [?]. This is the main reason why we will not consider these facts and only will review the most important results recently obtained. The next results survey some of the mentioned results. First of them is a fundamental result of Möhres [23]. Theorem 3.1 Let G be a group, all subgroups of which are subnormal. Then G is soluble. 5
The next result collects some results of Möhres [23] and Smith [29, 30, 31, 32]. Theorem 3.2 Let G be a group, all subgroups of which are subnormal. Then G is nilpotent in each one of the following cases: (1) if G is periodic and hypercentral, (2) if G is periodic and residually finite, (3) if G includes a nilpotent normal subgroup A such that G/A is bounded, (4) if G is periodic and residual nilpotent, or (5) if G is torsion-free. Further, in the paper [19], it is studied the nilpotency of those groups G with all subgroups subnormal and whose normal closures < x > G, x G satisfy some conditions. On the other hand, groups G in which the family L non sn satisfies the minimal condition, have been considered by Franciosi and de Giovanni [8]. These groups G with some additional restrictions become either Chernikov groups or groups with all subnormal subgroups. The dual study of groups G in which the family L non sn satisfies the maximal condition (or G satisfies Max (non-sn)) has appeared more successful. These groups have been considered by Kurdachenko and Smith [16] and we now state the basic results of that paper. Theorem 3.3 A locally nilpotent group G satisfies the condition L non sn if and only if every subgroup of G is subnormal. Theorem 3.4 A locally (soluble-by-finite) group G satisfies the condition L non sn if and only if G is a group of one of the following types: (1) G is polycyclic-by-finite, (2) every subgroup of G is subnormal, (3) the group G satisfies the following conditions: (3a) G B(G), (3b) G/B(G) is finitely generated, abelian-by-finite and torsion-free, (3c) B(G) is nilpotent, and (3d) for every element g B(G) each G invariant abelian factor of B(G) is finitely generated as Z < g > module. 6
Here and elsewhere B(G) denotes the Baer radical of the group G, that is the subgroup generated by all subnormal cyclic subgroups of G. A question related to the consideration of groups G that satisfy the condition Max (non-sn) is the study of the groups G, in which the family L non sn consists of finitely generated groups only. These groups have been considered by Heineken and Kurdachenko [10]. In what it concerns to the groups that satify weak chain conditions, the situation is the following. The groups G, in which the family L non sn satisfies the weak minimal condition (the groups with the condition Min (non-sn)), have been considered by Kurdachenko and Smith [?], whose results (similar to the ordinary minimal condition) we state now. Theorem 3.5 Suppose that the group G has an ascending series of subgroups, every factor of which either is locally nilpotent group or locally finite. If G satisfies the condition Min (non-sn), then either G is soluble by finite and minimax or every subgroup of G is subnormal. On the other hand, groups G in which the family L non sn satisfies the weak maximal condition (that is, groups with Max (non-sn)), were considered by Kurdachenko and Smith [18]. The situation here is rather complicated, but we can extract the simplest results of that paper. Theorem 3.6 Let G be a locally finite group. If G satisfies the condition Max (non-sn), then either G is a Chernikov group or every subgroup of G is subnormal. Theorem 3.7 Let G be a Baer group. If G satisfies the condition Max (non-sn), then every subgroup of G is subnormal. 4 Groups with rank restrictions on non subnormal subgroups A group G is said to have finite 0 rank r 0 (G) = r if G has a finite subnormal series with exactly r infinite cyclic factors, being the others periodic. We note that every refinement of one of these series has only r factors which are infinite cyclic; for, two finite subnormal series have isomorphic refinements. This allows to us to convince ourselves that the 0 rank is independent of the series. This numerical invariant is also known as the torsion-free rank of G. 7
Precedents of this concept are other invariants; for example, a polycyclicby-finite group has finite 0 rank which is exactly its Hirsch number. Note that if H is a subgroup of G, then r 0 (H) r 0 (G). Also, if H is a normal subgroup of G, then r 0 (G) = r 0 (H) + r 0 (G/H). In particular, if H is a periodic normal subgroup of G, then r 0 (G) = r0(g/h). In the sequel, if G is a group, we denote by P (G) the maximal normal periodic subgroup of G. The next results are taken from a paper of Kurdachenko and Smith [20]. Theorem 4.1 Let G be a soluble group whose non-subnormal subgroups have finite 0 rank. If G has infinite 0 rank, then G/P (G) is a torsion-free nilpotent group. Moreover, if P (G) has finite section rank, then G is a Baer group. We recall that a group G is said to have finite section p-rank r p (G) = r (p a prime), if the order of every elementary abelian p section of G is at most r. We say that G has finite section rank if r p (G) is finite for every prime p Other results contained in [20] are the following. Theorem 4.2 Suppose that G is a group whose non subnormal subgroups have finite section p rank, for some prime p, that includes a periodic subgroup of infinite p rank. Then (1) If G is locally (soluble-by-finite), then G is a soluble Baer group. (2) If G is a Baer group, then every subgroup of G is subnormal. Theorem 4.3 Let G be a soluble group whose non subnormal subgroups have finite section p rank for some prime p. If G has infinite section p rank but all periodic subgroups of G have finite section p rank, then G/O p (G) is a nilpotent group. Theorem 4.4 Let G be a soluble group whose non subnormal subgroups have finite section rank. If G has infinite section rank, then G is a Baer group. Following Robinson [25], we say that a group G has finite total rank r tot (G), if the sum r 0 (G) + p π(g) r p(g) = r tot (G) is finite. Using this invariant, Kurdachenko and Smith [20] were able to show the folloving results. 8
Theorem 4.5 Let G be a locally (soluble by finite) group whose non subnormal subgroups have finite total rank. If G includes a periodic subgroup of infinite total rank, then every subgroup of G is subnormal. Theorem 4.6 Let G be a soluble group whose non subnormal subgroups have finite total rank. If G has infinite total rank but all periodic subgroups have finite total rank, then G is a nilpotent group. Zaitsev [37] introduced the minimax rank of a group G. A group G is said to have finite minimax rank r mmx (G) = m, if every finite chain of subgroups < 1 >= G 0 G 1 G n = G in which all indexes G j+1 : G j are infinite has length n m and there exists a chain of subgroups as above with length exactly n = m. If such number n does not exists, then it is said that the minimax rank of G is infinite; obviously, if G is a finite group, we put r mmx (G) = 0. It is worth to noting that, in previous papers, Zaitsev himself used a rather different concept, namely the index of minimality, which replaced by the more adequated notion we mentioned above. In fact, it is shown that a locally (soluble-by-finite) group G has finite minimax rank if and only if it is minimax (that is G has a finite subnormal series, every factor of which satisfies Max or Min) (Zaitsev [36, 34]). Moreover, if every abelian subgroup of a radical group G has finite minimax rank then G is minimax (Baer [?], Zaitsev [35]). Note also that if H is a normal subgroup of a group G then r mmx (G) = r mmx (H) + r mmx (G/H). Also Kurdachenko and Smith [20] were able to show interesting results with this rank. Theorem 4.7 LetG be a soluble group whose non subnormal subgroups have finite minimax rank. If G has infinite minimax rank, then every subgroup of G is subnormal. Theorem 4.8 Let G be a soluble group whose non subnormal subgroups satisfy the minimal condition. If G is not a Chernikov group, then every subgroup of G is subnormal. 9
References [1] R. Baer. Situation der Untergruppen und Struktur der Gruppe. S.-B. Heidelberg Akad. 2 (1933), 12-17. [2] R. Baer. Polyminimaxgruppen. Math. Annalen 175 (1968), 1-43. [3] S. N. Chernikov. The groups with prescribed properties of family of infinite subgroups. Ukrain. Math. J. 19 (1967), 111-131. [4] S. N. Chernikov. Infinite non-abelian groups with minimal condition for non-invariant abelian subgroups. In The groups with restriction for subgroups. Nauk. dumka: Kiev - 1971, 106-115. [5] G. Cutolo. On groups satisfying the maximal condition on non-normal subgroups. Rivista Mat. pura ed applicata 91 (1991), 49-59. [6] G. Cutolo and L. A. Kurdachenko. Groups with a maximality condition for some non-normal subgroups. Geometriae Dedicata, 55 (1995), 279-292. [7] R. Dedekind. Über Gruppen deren sammtliche Teiler Normalteiler sind. Math. Annalen 48 (1897), 548-561. [8] S. Franciosi and F. de Giovanni. Groups satisfying the minimal condition on non-subnormal subgroups. In Infinite groups 1994 (Ravello), 63 72, W. de Gruyter, Berlin 1996. [9] B. Hartley. A note on the normalizer conditions. Proc. Cambridge Phil. Soc. 74 (1973), 11-15. [10] H. Heineken and L. A. Kurdachenko. Groups with subnormality for all subgroups that are not finitely generated. Annali Mat., 169 (1995), 203-232. [11] H. Heineken and I. J. Mohamed. A group with trivial center satisfying the normalizer condition. J. Algebra 10 (1968), 368-376. [12] H. Heineken and I. J. Mohamed. Groups with normalizer condition. Math. Annalen 198 (1972), 178-187. [13] H. Heineken and I. J. Mohamed. Non-nilpotent groups with normalizer condition. Lecture Notes Math. 372 (1974), 357-360. 10
[14] L. A. Kurdachenko and V. E. Gorezky. On groups with the weak minimal and maximal conditions for non-normal subgroups. Ukrain. Math. J.41 (1989), 17O5-17O9. [15] L. A. Kurdachenko and V. V. Pylaev. On groups dual to Dedekind. Dokl. AN Ukrain.SSR, ser. A, 10 ( 1989), 21-22. [16] L. A. Kurdachenko and H. Smith. Groups with the maximal condition on non subnormal subgroups. Bolletino Unione Mat. Ital. 10 B (1996), 441-460. [17] L. A. Kurdachenko and H. Smith. Groups with the weak minimal condition for non subnormal subgroups. Annali Mat. 173 (1997), 299-312. [18] L. A. Kurdachenko and H. Smith. Groups with the weak maximal condition for non subnormal subgroups. Ricerche Mat. 47 (1998), 29-49. [19] L. A. Kurdachenko and H. Smith. The nilpotence of some groups with all subgroups subnormal. Publicaciones Mat. 42 (1998), 411-421. [20] L. A. Kurdachenko and H. Smith. Groups with the rank restriction for non subnormal subgroups, to appear [21] F. Menegazzo. Groups of Heineken - Mohamed. J. Algebra. 171 (1995), 807-825. [22] G. A. Miller and H. Moreno. Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 4 (1903), 389-404. [23] W. Möhres. Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind. Archiv Math. 54 (1990), 232-235. [24] A. Yu. Ol shanskij. Geometry of defining relations in groups. Kluwer Acad. Publ., Dordrecht 1991. [25] D. J. S. Robinson. Finiteness conditions and generalized soluble groups. Springer Verlag, Berlin 1972. [26] O. Yu. Schmidt. The groups, all subgroups of which are special. Math. Sbornik 31 (1924), 366-372. [27] O. Yu. Schmidt. The groups having only one ckass of non-invariant subgroups. Math. Sbornik 33 (1926), 161-172. 11
[28] O. Yu. Schmidt. The groups with two classes of non-invariant subgroups The works of a seminar on Group Theory, 1938, 7-26. [29] H. Smith. Residually finite groups with all subgroups subnormal. Bull. London Math. Soc.31 (1999), 679-680. [30] H. Smith. Nilpotent-by-(finite exponent) groups with all subgroups subnormal. J. Group Theory 3 (2000), 47-56. [31] H. Smith. Residually nilpotent groups with all subgroups subnormal. J. Algebra 244 (2001), 845-850. [32] H. Smith. Torsion-free groups with all subgroups subnormal. Archiv Math. 76 (2001),, 1-6. [33] V. P. Shunkov. On locally finite groups with minimal condition for abelian subgroups. Algebra and logic 9 (1970), 579-615. [34] D. I. Zaitsev. Groups satisfying the weak minimal condition. Ukrain. Math. J. 20 (1968), 472-482. [35] D. I. Zaitsev. On groups satisfying the weak minimal condition. Mat. Sbornik 78 (1969), 323-331. [36] D. I. Zaitsev. The theory of minimax groups. Ukrain. Math. J. 23 (1971), 652-660. [37] D. I. Zaitsev. On the index of minimality of the groups. In Investigations of the groups on prescribed properties of subgroups, Math. Inst. Kiev - 1974, 72-130. 12