Groups whose non-normal subgroups have small commutator subgroup

The structure of groups whose non-normal subgroups have a finite commutator subgroup is investigated. In particular, it is proved that if k is a positive integer and G is a locally graded group in which every non-normal subgroup has finite commutator subgroup of order at most k, then the commutator...

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Дата:	2007
Автори:	De Falco, M., de Giovanni, F., Musella, C.
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Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2007
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	Groups whose non-normal subgroups have small commutator subgroup / M. De Falco, F. de Giovanni, C. Musella // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 3. — С. 46–58. — Бібліогр.: 16 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1523712019-06-11T01:25:23Z Groups whose non-normal subgroups have small commutator subgroup De Falco, M. de Giovanni, F. Musella, C. The structure of groups whose non-normal subgroups have a finite commutator subgroup is investigated. In particular, it is proved that if k is a positive integer and G is a locally graded group in which every non-normal subgroup has finite commutator subgroup of order at most k, then the commutator subgroup of G is finite. Moreover, groups with finitely many normalizers of subgroups with large commutator subgroup are studied. 2007 Article Groups whose non-normal subgroups have small commutator subgroup / M. De Falco, F. de Giovanni, C. Musella // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 3. — С. 46–58. — Бібліогр.: 16 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20F24. http://dspace.nbuv.gov.ua/handle/123456789/152371 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The structure of groups whose non-normal subgroups have a finite commutator subgroup is investigated. In particular, it is proved that if k is a positive integer and G is a locally graded group in which every non-normal subgroup has finite commutator subgroup of order at most k, then the commutator subgroup of G is finite. Moreover, groups with finitely many normalizers of subgroups with large commutator subgroup are studied.
format	Article
author	De Falco, M. de Giovanni, F. Musella, C.
spellingShingle	De Falco, M. de Giovanni, F. Musella, C. Groups whose non-normal subgroups have small commutator subgroup Algebra and Discrete Mathematics
author_facet	De Falco, M. de Giovanni, F. Musella, C.
author_sort	De Falco, M.
title	Groups whose non-normal subgroups have small commutator subgroup
title_short	Groups whose non-normal subgroups have small commutator subgroup
title_full	Groups whose non-normal subgroups have small commutator subgroup
title_fullStr	Groups whose non-normal subgroups have small commutator subgroup
title_full_unstemmed	Groups whose non-normal subgroups have small commutator subgroup
title_sort	groups whose non-normal subgroups have small commutator subgroup
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2007
url	http://dspace.nbuv.gov.ua/handle/123456789/152371
citation_txt	Groups whose non-normal subgroups have small commutator subgroup / M. De Falco, F. de Giovanni, C. Musella // Algebra and Discrete Mathematics. — 2007. — Vol. 6, № 3. — С. 46–58. — Бібліогр.: 16 назв. — англ.
series	Algebra and Discrete Mathematics
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first_indexed	2025-07-13T02:56:14Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2007). pp. 46 – 58 c© Journal “Algebra and Discrete Mathematics” Groups whose non-normal subgroups have small commutator subgroup M. De Falco, F. de Giovanni and C. Musella Communicated by D. Simson Abstract. The structure of groups whose non-normal sub- groups have a finite commutator subgroup is investigated. In par- ticular, it is proved that if k is a positive integer and G is a lo- cally graded group in which every non-normal subgroup has finite commutator subgroup of order at most k, then the commutator subgroup of G is finite. Moreover, groups with finitely many nor- malizers of subgroups with large commutator subgroup are studied. Introduction It is well known that a group has only normal subgroups (i.e. is a Dedekind group) if and only if it is either abelian or the direct product of a quaternion group of order 8 and a periodic abelian group with no ele- ments of order 4. The structure of groups for which the set of non-normal subgroups is small in some sense has been studied by many authors in several different situations. A group G is called metahamiltonian if every non-normal subgroup of G is abelian. In a series of relevant papers, G.M. Romalis and N.F. Sesekin investigated the behaviour of (generalized) sol- uble metahamiltonian groups and proved in particular that such groups have finite commutator subgroup (see [14],[15],[16]). However, since every group with commutator subgroup of prime order is obviously metahamil- tonian, there is no bound for the order of the commutator subgroup of soluble metahamiltonian groups. The aim of this paper is to study groups in which every subgroup either is normal or has finite commutator subgroup. The first step in this 2000 Mathematics Subject Classification: 20F24. Key words and phrases: normal subgroup, commutator subgroup. Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 47 direction is of course the case of groups whose proper subgroups have a fi- nite commutator subgroup; such groups were considered by V.V. Belyaev and N.F. Sesekin (see [1],[2]) and their results were later improved by B. Bruno and R.E. Phillips [3]. In order to avoid Tarski groups (i.e. infi- nite simple groups whose proper non-trivial subgroups have prime order) and other similar pathologies, we will restrict our attention to locally graded groups. Recall here that a group G is said to be locally graded if every finitely generated non-trivial subgroup of G contains a proper subgroup of finite index; of course, all locally (soluble-by-finite) groups are locally graded. We will prove that if k is a positive integer and G is a locally graded group in which every non-normal subgroup has finite com- mutator of order at most k, then the commutator subgroup of G is finite. A similar result holds if the restriction on commutator subgroups is im- posed only to infinite subgroups, provided that G is not a Černikov group. Note that the case k = 1 of our theorem is precisely the above quoted result by Romalis and Sesekin. If G is any group whose commutator sub- group is of type p∞ for some prime number p, all non-normal subgroups of G have a finite commutator subgroup; thus the bound assumption in the previous statement cannot be removed. Finally, in the last section we extend our results to groups with finitely many normalizers of subgroups with large commutator subgroup. Most of our notation is standard and can be found in [13]. 1. Groups with small non-normal subgroups In the investigation concerning the structure of groups whose non-normal subgroups are small in some sense, one has to consider the case of groups in which every subgroup of finite index is normal. Of course, in this situation information can be obtained only on the factor group with re- spect to the finite residual. Recall that the finite residual of a group G is the intersection of all (normal) subgroups of finite index of G, and G is residually finite if its finite residual is trivial; residually finite groups are obviously locally graded. It is easy to show that every periodic residually finite group whose finite homomorphic images are Dedekind groups is likewise a Dedekind group. The situation is different in the case of non-periodic groups; in fact, the direct product Q8 × Q2 (where Q8 is the quaternion group of order 8 and Q2 is the additive group of rational numbers whose denomi- nators are powers of 2) is a residually finite non-abelian group and all its subgroups of finite index are normal. Our first lemma describes residually finite groups whose finite homomorphic images are Dedekind groups. Jo u rn al A lg eb ra D is cr et e M at h .48 Small commutator subgroup Lemma 1. Let G be a non-abelian residually finite group. Then all sub- groups of finite index of G are normal if and only if Z(G) = G′ × A, where \|G′\| = 2, \|G/Z(G)\| = 4, A2 = A4 and G/A2 is a Dedekind group. Proof. Suppose first that every subgroup of finite index of G is normal, and let L be the set of all subgroups of finite index of G. If H is any element of L, the factor group G/H is a Dedekind group an so G′H/H has order at most 2. As ⋂ H∈L H = {1}, it follows that G′ has exponent 2. Let X be any finite subgroup of G′, and let H be a subgroup of finite index of G such that X ∩ H = {1}; then X ≃ XH/H ≤ (G/H)′ and hence \|X\| ≤ 2. Therefore G′ has order 2. By hypothesis, the group G contains a normal subgroup A such that G/A is a quaternion group of order 8. Clearly, A ∩ G′ = {1}, so that A ≤ Z(G) and Z(G) = G′ × A; in particular \|G/Z(G)\| = 4. The factor group G/A4 is periodic and residually finite, so that it is a Dedekind group; as A ∩ G′ = {1}, it follows that A/A4 has exponent 2. Therefore A2 = A4 and G/A2 is a Dedekind group. Conversely, suppose that G has the structure described in the state- ment, and let X be any subgroup of finite index of G. Since A2 = A4, the subgroup X ∩A2 has odd index in A2; moreover, G/A2 is a Dedekind 2-group, so that G/X ∩ A2 is likewise a Dedekind group, and hence the subgroup X is normal in G. The consideration of the direct product Q8×Q8 proves that the above lemma cannot be extended to the case of groups having a residual system whose factors are finite Dedekind groups. Our next result deals with locally graded groups in which every non- normal subgroup has locally finite commutator subgroup. Lemma 2. Let G be a locally graded group whose non-normal subgroups have a locally finite commutator subgroup. Then the commutator sub- group G′ of G is locally finite. Proof. Let T be the largest normal subgroup of G which is locally finite, and consider any subgroup X of G such that X ′ is not locally finite. Then X is normal in G and G/X is a Dedekind group, so that in particular G′′ ≤ X. It follows that every proper subgroup of G′′ has locally finite commutator subgroup. If G′′ is a finitely generated non-trivial group, it Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 49 contains a proper normal subgroup K of finite index; then K ′ is a locally finite subnormal subgroup of G, so that K ′ is contained in T . On the other hand, if G′′ is not finitely generated, its commutator subgroup G(3) is locally finite and hence G(3) ≤ T . Therefore G/T is a soluble-by-finite group in any case, and replacing G by G/T it can be assumed without loss of generality that G is a soluble-by-finite group. As the hypotheses are inherited by subgroups, in order to prove the statement we may also suppose that G is finitely generated and by a further reduction that it contains no locally finite non-trivial normal sub- groups. Assume for a contradiction that G is not torsion-free. Then G is not nilpotent, so that it has a finite non-nilpotent homomorphic image (see [13] Part 2, Theorem 10.51) and in particular G contains a non- normal subgroup H such that the index \|G : H\| is finite. As the sub- group H ′ is locally finite, also the core A = HG of H in G has locally finite commutator subgroup. Thus A′ = {1}, so that A is an abelian normal subgroup of G such that G/A is finite, and in particular A is torsion- free and finitely generated. For each positive integer n, the subgroup An has finite index in G and so its centralizer CG(An) is central-by-finite; it follows from Schur’s theorem that the commutator subgroup CG(An)′ is finite, and hence CG(An) is torsion-free abelian. Let x 6= 1 be an element of finite order of G; then the subgroup 〈x, An〉 is not abelian and 〈x, An〉′ is torsion-free, so that 〈x, An〉 must be normal in G. Therefore 〈x〉 = ⋂ n>0 〈x, An〉 is likewise a normal subgroup of G and hence [A, x] = {1}. This contra- diction shows that G is torsion-free, so that all its non-normal subgroups are abelian and hence G itself is abelian. The lemma is proved. The following lemma shows that we actually work within the universe of finite-by-soluble groups. Lemma 3. Let G be a locally graded group whose non-normal subgroups have a finite commutator subgroup. Then either G is soluble or the sub- group G(3) is finite. Proof. Assume that G is not soluble, and let L be the set of all subgroups of G with infinite commutator subgroup. If X is any element of L, then X is normal in G and G/X is a Dedekind group; it follows in particular that M = ⋂ X∈L X Jo u rn al A lg eb ra D is cr et e M at h .50 Small commutator subgroup is a normal subgroup of G and G′′ is contained in M . Moreover, ev- ery proper subgroup of M has finite commutator subgroup and hence either M ′ is finite or M is a Černikov group (see [3], Theorem 2). In order to prove that G(3) is finite, we may obviously suppose that M is a Černikov group. Let J be the largest divisible subgroup of M , and let x be any element of M . As the subgroup 〈x, J〉 is soluble, it is properly contained in M and hence 〈x, J〉′ is finite. Thus x has finitely many con- jugates in 〈x, J〉 and so also in M ; it follows from Dietzmann’s lemma that M = EJ , where E is a finite normal subgroup of M . Therefore M ′ is finite and the lemma is proved. For our purposes, we also need the following lemma, which is a special case of a result concerning groups whose non-normal subgroups have finite conjugacy classes (see [9], Proposition 4.1). Lemma 4. Let G be an abelian-by-finite group whose non-normal sub- groups have a finite commutator subgroup. Then the commutator sub- group G′ of G is a Černikov group. Corollary 1. Let G be a residually finite group whose non-normal sub- groups have a finite commutator subgroup. Then the commutator sub- group G′ of G is finite. Proof. If every subgroup of finite index of G is normal, we have that all finite homomorphic images of G are Dedekind groups and it follows from Lemma 1 that G′ has order at most 2. Suppose now that G contains a non-normal subgroup X of finite index; then X ′ is finite and hence also the core XG of X has finite commutator subgroup. As the factor group G/X ′ G is abelian-by-finite, its commutator subgroup G′/X ′ G is a Černikov group by Lemma 4. Therefore G′ is likewise a Černikov group, and hence it is finite. Lemma 5. Let G be a locally graded group whose non-normal subgroups have a finite commutator subgroup. Then either G is locally nilpotent or every finitely generated subgroup of G has finite commutator subgroup. Proof. Assume that G contains a finitely generated non-nilpotent sub- group H, and let E be any finitely generated subgroup of G. As G is soluble-by-finite by Lemma 3, the subgroup K = 〈E, H〉 has a finite non-nilpotent homomorphic image (see [13] Part 2, Theorem 10.51) and in particular K contains a non-normal subgroup X such that the index \|K : X\| is finite. By hypothesis the commutator subgroup X ′ of X is finite, and hence K is finite-by-abelian-by-finite. As finitely generated Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 51 abelian-by-finite groups are residually finite, it follows from Lemma 1 that K ′ is finite, so that E′ is likewise finite and the lemma is proved. Corollary 2. Let G be a locally graded group whose non-normal sub- groups have a finite commutator subgroup. Then G locally satisfies the maximal condition on subgroups. We can now prove the main result of this section. Theorem 1. Let k be a positive integer, and let G be a locally graded group whose infinite non-normal subgroups have a finite commutator sub- group of order at most k. Then either G is a Černikov group or its commutator subgroup G′ is finite. Proof. Clearly, every non-normal subgroup of G has finite commutator subgroup, so that in particular G′ is locally finite by Lemma 2; more- over, G locally satisfies the maximal condition on subgroups by Corol- lary 2 and hence every finitely generated subgroup of G has finite com- mutator subgroup. It follows from Lemma 3 that G is finite-by-soluble, so that without loss of generality we may suppose that G is a soluble group. Assume for a contradiction that G is neither finite-by-abelian nor a Černikov group, so that in particular G is not finitely generated. Let E be any infinite finitely generated subgroup of G, and suppose that \|E′\| > k; then E is normal in G and G/E is a Dedekind group, so that \|G′E/E\| ≤ 2. Moreover, E satisfies the maximal condition on subgroups, so that E∩G′ is finite, and hence G′ is likewise finite. This contradiction shows that \|E′\| ≤ k for each infinite finitely generated subgroup E of G. On the other hand, as G′ is infinite, there exists a finite subgroup X of G such that \|X ′\| > k, and every finitely generated subgroup of G containing X must be finite. Therefore the group G is locally finite. Assume that the normal closure N = XG of X is infinite, and let H1 ≥ H2 ≥ . . . ≥ Hn ≥ Hn+1 ≥ . . . be any sequence of infinite normal subgroups of N . Since every infinite subgroup containing X is normal in G, we have that all proper subgroups of N containing X are finite, and so N = HnX for each n; if m is a positive integer such that Hn∩X = Hm∩X for all n ≥ m, it follows that Hn = Hm for all n ≥ m. Therefore the group N satisfies the minimal condition on normal subgroups. Let i be the largest positive integer such that the i-th term N (i) of the derived series of N is infinite. Then N (i+1) is finite and N = XN (i), so that the index \|N : N (i)\| is finite and hence N is finite-by-abelian-by-finite. As the minimal condition on Jo u rn al A lg eb ra D is cr et e M at h .52 Small commutator subgroup normal subgroups is inherited by subgroups of finite index (see [13] Part 1, Theorem 5.21), it follows that N is a Černikov group. Then G/CG(N) is likewise a Černikov group (see [13] Part 1, Theorem 3.29), so that CG(N) is not a Černikov group and hence there exists an abelian subgroup A of CG(N) which does not satisfy the minimal condition on subgroups (see [13] Part 1, Theorem 3.32). In particular, A contains a subgroup of the form B = B1 ×B2, where both B1 and B2 are infinite and B ∩X = {1}. Clearly, the subgroups B1X and B2X are normal in G, so that also X = B1X ∩B2X is normal in G and this contradiction shows that XG is finite. On the other hand, every infinite subgroup of G/XG is normal and hence G/XG is either a Černikov group or a Dedekind group (see [4]). Therefore either G is a Černikov group or its commutator subgroup G′ is finite, and this last contradiction completes the proof of the theorem. The description of locally graded group whose non-normal subgroups have boundedly finite commutator subgroups is an easy consequence of the above result. Theorem 2. Let k be a positive integer, and let G be a locally graded group whose non-normal subgroups have a finite commutator subgroup of order at most k. Then the commutator subgroup G′ of G is finite. Proof. Assume for a contradiction that G′ is infinite. It follows from The- orem 1 that G is a Černikov group; in particular, G is locally finite and hence there exists a finite subgroup E of G such that \|E′\| > k. On the other hand, by hypothesis E is normal in G and G/E is a Dedekind group, so that G′ is finite, a contradiction. 2. Groups with few normalizers of large subgroups In a famous paper of 1955, B.H. Neumann [11] proved that each subgroup of a group G has finitely many conjugates if and only if the centre Z(G) has finite index, and hence central-by-finite groups are precisely those groups in which the normalizers of subgroups have finite index. This result suggests that the behaviour of normalizers has a strong influence on the structure of the group. In fact, it follows easily from a result of Y.D. Polovickĭı [12] that a group has finitely many normalizers of abelian subgroups if and only if it is central-by-finite. More recently, groups with finitely many normalizers of χ-subgroups have been studied for various choices of the property χ (see for instance [5],[6],[7],[8]). The aim of this section is to study groups with finitely many normalizers of subgroups with infinite commutator subgroup; the first step of this investigation is Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 53 of course the case of groups whose non-normal subgroups have a finite commutator subgroup. It follows in particular from Lemma 1 that every residually finite group whose subgroups of finite index are normal is central-by-finite; this latter property actually characterizes residually finite groups with finitely many normalizers of subgroups of finite index. In fact, we have: Lemma 6. Let G be a residually finite group with finitely many nor- malizers of subgroups of finite index. Then the factor group G/Z(G) is finite. Proof. Let NG(X1), . . . , NG(Xm) be the proper normalizers of subgroups of finite index of G. The intersection N = m⋂ i=1 NG(Xi) normalizes each subgroup of finite index of G and the index \|G : N \| = r is obviously finite. If Ḡ is any finite homomorphic image of G, it follows that each subgroup of Ḡ has at most r conjugates and so the order of Ḡ′ is bounded by a positive integer s = s(r) depending only on r (see [10] and [13] Part 1, p.102). Let L be the set of all normal subgroups of finite index of G. Then (G′)s! ≤ ⋂ H∈L H = {1} and in particular G′ is periodic. Moreover, all subgroups of finite index of the residually finite group N are normal, and hence N/Z(N) is finite by Lemma 1; thus G is abelian-by-finite and G′ is locally finite. As- sume for a contradiction that G′ is infinite, so that it contains a finite subgroup E with \|E\| > r. As G is residually finite, there exists a nor- mal subgroup H of G such that G/H is finite and H ∩ E = {1}; then the commutator subgroup of G/H contains more than r elements, and this contradiction shows that G′ must be finite. Finally, it is clear that any residually finite group with finite commutator subgroup is central- by-finite. Next lemma is an extension of Corollary 1. Lemma 7. Let G be a residually finite group with finitely many normaliz- ers of subgroups with infinite commutator subgroup. Then the commutator subgroup G′ of G is finite. Jo u rn al A lg eb ra D is cr et e M at h .54 Small commutator subgroup Proof. Assume that the statement is false, and choose a counterexam- ple G such that the set {NG(X1), . . . , NG(Xm)}, consisting of all proper normalizers of subgroups with infinite commutator subgroup, has smallest order m. Then m > 0 by Corollary 1, and for each i = 1, . . . , m the group NG(Xi) has less than m proper normalizers of subgroups with infinite commutator subgroup. It follows that NG(Xi) has finite commutator subgroup, and this contradiction proves the statement. It is quite easy to prove the following result, which improves Theo- rem 2. Theorem 3. Let k be a positive integer, and let G be a locally graded group with finitely many normalizers of subgroups whose commutator sub- group has more than k elements. Then the commutator subgroup G′ of G is finite. Proof. Assume that the statement is false, and choose a counterexam- ple G such that the set {NG(X1), . . . , NG(Xm)}, consisting of all proper normalizers of subgroups of G whose commutator subgroup contains more than k elements, has smallest order m. Of course, m > 0 by Theorem 2, and for every i = 1, . . . , m the group NG(Xi) has fewer than m proper normalizers of subgroups whose commutator subgroup contains more than k elements, so that NG(Xi) ′ is finite. Since the subgroup NG(Xi) has obviously finitely many conjugates in G, also the conjugacy class of NG(Xi) ′ is finite, and hence the normal subgroup K = 〈NG(X1) ′, . . . , NG(Xm)′〉G is finite by Dietzmann’s lemma; in particular, the factor group G/K is likewise locally graded. Moreover, every subgroup of G/K whose com- mutator subgroup has more than k elements must be normal and so G/K has finite commutator subgroup by Theorem 2. Thus also G′ is finite, and this contradiction proves the theorem. In order to generalize Theorem 1 to the case of groups with finitely many normalizers of infinite subgroups with unbounded commutator sub- groups, we need some more lemmas. Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 55 Lemma 8. Let G be a group having finitely many normalizers of sub- groups with infinite commutator subgroup. Then G contains a character- istic subgroup M of finite index such that NM (X) is normal in M for each subgroup X with infinite commutator subgroup. Proof. If X is any subgroup of G with infinite commutator subgroup, the normalizer NG(X) has obviously finitely many images under auto- morphisms of G; in particular, the index \|G : NG(NG(X))\| is finite. It follows that also the characteristic subgroup M(X) = ⋂ α∈Aut G NG(NG(X))α has finite index in G. Let H be the set of all subgroups of G with infinite commutator subgroup. If X and Y are elements of H such that NG(X) = NG(Y ), then M(X) = M(Y ), and hence also M = ⋂ X∈H M(X) is a characteristic subgroup of finite index of G. Let X be any subgroup of M such that X ′ is infinite. Then M ≤ M(X) ≤ NG(NG(X)), and so the normalizer NM (X) = NG(X)∩M is a normal subgroup of M . Lemma 9. Let G be a soluble-by-finite group having finitely many nor- malizers of subgroups with infinite commutator subgroup. Then G locally satisfies the maximal condition on subgroups. Proof. As the hypotheses are inherited by subgroups, it can be assumed without loss of generality that the group G is finitely generated. It follows from Lemma 8 that G contains a normal subgroup M such that G/M is finite and NM (X) is normal in M for each subgroup X of M with infinite commutator subgroup. Suppose first that every subgroup of finite index of M is subnormal. Then all finite homomorhic images of M are nilpotent, so that G is nilpotent-by-finite (see [13] Part 2, Theorem 10.51) and hence it satisfies the maximal condition on subgroups. On the other hand, if M contains a non-subnormal subgroup X of finite index, the commutator subgroup X ′ must be finite; then G is finite-by-abelian-by-finite and so it satisfies the maximal condition on subgroups also in this case. Jo u rn al A lg eb ra D is cr et e M at h .56 Small commutator subgroup Lemma 10. Let G be a group having finitely many normalizers of sub- groups with infinite commutator subgroup. If G contains a soluble-by- finite normal subgroup N such that N ′ is infinite, then G is likewise soluble-by-finite. Proof. As N ′ is infinite, the factor group G/N has only finitely many normalizers of subgroups, and hence it is central-by-finite by Polovickĭı’s theorem. Therefore G is soluble-by-finite. Theorem 4. Let k be a positive integer, and let G be a locally graded group with finitely many normalizers of infinite subgroups whose commu- tator subgroup has more than k elements. Then either G is a Černikov group or its commutator subgroup G′ is finite. Proof. Assume that the statement is false, and choose a counterexam- ple G such that the set {NG(X1), . . . , NG(Xm)}, consisting of all proper normalizers of infinite subgroups of G whose com- mutator subgroup contains more than k elements, has smallest order m. Of course, m > 0 by Theorem 1, and for every i = 1, . . . , m the group NG(Xi) has less than m proper normalizers of infinite subgroups whose commutator subgroup has more than k elements, so that either NG(Xi) ′ is finite or NG(Xi) is a Černikov group. Suppose that NG(Xi) ′ is finite for some i. The factor group G/(NG(Xi) ′)G has fewer than m proper nor- malizers of infinite subgroups whose commutator subgroup contains more than k elements, and hence G/(NG(Xi) ′)G is either finite-by-abelian or a Černikov group. On the other hand, the subgroup NG(Xi) ′ has finitely many conjugates in G and so its normal closure (NG(Xi) ′)G is finite by Dietzmann’s lemma; this is a contradiction, since G is a counterexam- ple to the statement. Therefore NG(Xi) is a Černikov group for each i = 1, . . . , m and in particular all non-normal subgroups of G have lo- cally finite commutator subgroup, so that G′ is locally finite by Lemma 2. Moreover, the index \|G : NG(Xi)\| must be infinite for every i; it follows that NG(NG(Xi)) 6= NG(Xj) for all i and j, and so every NG(Xi) is a normal subgroup of G. As G/NG(Xi) has less than m proper normalizers of infinite subgroups whose commutator subgroup contains more than k elements, the group G is soluble-by-finite by Lemma 10, and hence it locally satisfies the maximal condition on subgroups by Lemma 9. Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 57 The factor group G/CG(NG(Xi)) is isomorphic to a group of auto- morphisms of the Černikov group NG(Xi) and obviously CG(NG(Xi)) ≤ NG(Xi), so that G/CG(NG(Xi)) cannot be periodic (see [13] Part 1, The- orem 3.29) and in particular NG(X1) ∪ . . . ∪ NG(Xm) is a proper subset of G. Let x be an element of the set G \ m⋃ i=1 NG(Xi), so that the commutator subgroup of every infinite non-normal subgroup of G containing x has order at most k. As G′ is infinite, there exists a finitely generated subgroup E of G such that \|E′\| > k, and so 〈x, E〉 is an infinite group whose commutator subgroup contains more than k elements. Therefore 〈x, E〉 is a normal subgroup of G and G/〈x, E〉 is a Dedekind group, so that G′/G′ ∩ 〈x, E〉 is finite. Finally, 〈x, E〉 satisfies the maximal condition on subgroups, so that G′ ∩ 〈x, E〉 is finite and hence G′ is likewise finite. This last contradiction completes the proof of the theorem. References [1] V.V. Belyaev, Groups of Miller-Moreno type, Sibirsk. Mat. Ž. 19 (1978), 509–514. [2] V.V. Belyaev, N.F. Sesekin, Infinite groups of Miller-Moreno type, Acta Math. Acad. Sci. Hungar. 26 (1975), 369–376. [3] B. Bruno, R.E. Phillips, On minimal conditions related to Miller-Moreno type groups, Rend. Sem. Mat. Univ. Padova 69 (1983), 153–168. [4] S.N. Černikov, Groups with given properties of systems of infinite subgroups, Ukrain. Math. J. 19 (1967), 715–731. [5] M. De Falco, F. de Giovanni, C. Musella, Groups with finitely many normalizers of non-polycyclic subgroups, Algebra Colloq., to appear. [6] F. De Mari, F. de Giovanni Groups with finitely many normalizers of non-abelian subgroups, Ricerche Mat. 55 (2006), 311-317. [7] F. De Mari, F. de Giovanni, Groups with finitely many normalizers of subnormal subgroups, J. Algebra 304 (2006), 382–396. [8] F. De Mari, F. de Giovanni, Groups with finitely many normalizers of non- nilpotent subgroups, Math. Proc. Roy. Irish Acad., 107A (2007), 143–152. [9] L.A. Kurdachenko, J. Otal, A. Russo, G. Vincenzi, Groups whose non-normal subgroups have finite conjugacy classes, Math. Proc. Roy. Irish Acad. 104A (2004), 177–189. [10] I.D. Macdonald, Some explicit bounds in groups with finite derived groups, Proc. London Math. Soc. 11 (1961), 23–56. Jo u rn al A lg eb ra D is cr et e M at h .58 Small commutator subgroup [11] B.H. Neumann, Groups with finite classes of conjugate subgroups, Math. Z. 63 (1955), 76–96. [12] Y.D. Polovickĭı, Groups with finite classes of conjugate infinite abelian subgroups, Soviet Math. (Iz. VUZ) 24 (1980), 52–59. [13] D.J.S. Robinson Finiteness Conditions and Generalized Soluble Groups, Springer, 1972. [14] G.M. Romalis, N.F. Sesekin, Metahamiltonian groups, Ural. Gos. Univ. Mat. Zap. 5 (1966), 101–106. [15] G.M. Romalis, N.F. Sesekin, Metahamiltonian groups II, Ural. Gos. Univ. Mat. Zap. 6 (1968), 52–58. [16] G.M. Romalis, N.F. Sesekin, Metahamiltonian groups III, Ural. Gos. Univ. Mat. Zap. 7 (1969/70), 195–199. Contact information M. De Falco Dipartimento di Matematica e Applicazioni, via Cintia, I - 80126 Napoli (Italy) E-Mail: mdefalco@unina.it F. de Giovanni Dipartimento di Matematica e Applicazioni, via Cintia, I - 80126 Napoli (Italy) E-Mail: degiovan@unina.it C. Musella Dipartimento di Matematica e Applicazioni, via Cintia, I - 80126 Napoli (Italy) E-Mail: cmusella@unina.it Received by the editors: 26.06.2007 and in final form 28.01.2008.

Groups whose non-normal subgroups have small commutator subgroup

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