Groups with many self-normalizing subgroups

This paper investigates the structure of groups in which all members of a given relevant set of subgroups are selfnormalizing. In particular, soluble groups in which every nonabelian (or every infinite non-abelian) subgroup is self-normalizing are described.

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Hauptverfasser:	M. De Falco, F. de Giovanni, Musella, C.
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spelling	irk-123456789-1570212019-06-20T01:27:47Z Groups with many self-normalizing subgroups M. De Falco F. de Giovanni Musella, C. This paper investigates the structure of groups in which all members of a given relevant set of subgroups are selfnormalizing. In particular, soluble groups in which every nonabelian (or every infinite non-abelian) subgroup is self-normalizing are described. 2009 Article Groups with many self-normalizing subgroups / M. De Falco, F. de Giovanni, C. Musella // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 55–65. — Бібліогр.: 10 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20F19. http://dspace.nbuv.gov.ua/handle/123456789/157021 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	This paper investigates the structure of groups in which all members of a given relevant set of subgroups are selfnormalizing. In particular, soluble groups in which every nonabelian (or every infinite non-abelian) subgroup is self-normalizing are described.
format	Article
author	M. De Falco F. de Giovanni Musella, C.
spellingShingle	M. De Falco F. de Giovanni Musella, C. Groups with many self-normalizing subgroups Algebra and Discrete Mathematics
author_facet	M. De Falco F. de Giovanni Musella, C.
author_sort	M. De Falco
title	Groups with many self-normalizing subgroups
title_short	Groups with many self-normalizing subgroups
title_full	Groups with many self-normalizing subgroups
title_fullStr	Groups with many self-normalizing subgroups
title_full_unstemmed	Groups with many self-normalizing subgroups
title_sort	groups with many self-normalizing subgroups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/157021
citation_txt	Groups with many self-normalizing subgroups / M. De Falco, F. de Giovanni, C. Musella // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 55–65. — Бібліогр.: 10 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT mdefalco groupswithmanyselfnormalizingsubgroups AT fdegiovanni groupswithmanyselfnormalizingsubgroups AT musellac groupswithmanyselfnormalizingsubgroups
first_indexed	2025-07-14T09:21:56Z
last_indexed	2025-07-14T09:21:56Z
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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 4. (2009). pp. 55 – 65 c⃝ Journal “Algebra and Discrete Mathematics” Groups with many self-normalizing subgroups M. De Falco, F. de Giovanni and C. Musella Communicated by I. Ya. Subbotin Dedicated to Leonid A. Kurdachenko on his sixtieth birthday Abstract. This paper investigates the structure of groups in which all members of a given relevant set of subgroups are self- normalizing. In particular, soluble groups in which every non- abelian (or every infinite non-abelian) subgroup is self-normalizing are described. Introduction It is well known that a group has only normal subgroups 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 elements 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; in particular, metahamiltonian groups (i.e. groups whose non-abelian subgroups are normal) were introduced and investigated in a series of papers by G.M. Romalis and N.F. Sesekin (see [8],[9],[10]); further infor- mations on the structure of such groups have been later obtained by N.F. Kuzennyi and N.N. Semko [4]. Dualizing this point of view, we are interested here to (generalized) soluble groups in which all members of a given relevant set of subgroups are self-normalizing. Of course, any group whose non-trivial subgroups are self-normalizing is periodic and simple, and it is elementary to show that locally finite groups with such property either are trivial or have prime order. The consideration of Tarski groups (i.e. infinite simple 2000 Mathematics Subject Classification: 20F19. Key words and phrases: self-normalizing subgroup, minimal non-abelian group. Jo u rn al A lg eb ra D is cr et e M at h .56 Self-normalizing subgroups groups whose proper non-trivial subgroups have prime order) shows that there exist infinite groups in which every non-trivial subgroup is self- normalizing; such infinite simple groups can have even a more complicated structure, since it follows from a result of V.N. Obraztsov [5] that any Tarski group can be properly embedded in a simple group in which every proper subgroup either has prime order or is a Tarski group. As in the case of metahamiltonian groups, we look at the set of all non-abelian subgroups of a group. The structure of (soluble) groups in which every non-abelian subgroup is self-normalizing will be described in Section 2, while the final Section 3 will be devoted to the study of groups whose infinite non-abelian subgroups are self-normalizing. Let ℋ be the class of all groups in which every non-abelian subgroup is self-normalizing. Clearly, all groups whose proper subgroups are abelian belong to the class ℋ, and we shall denote by ℋ∗ the class of ℋ-groups containing proper non-abelian subgroups; it will be proved that the struc- ture of soluble non-nilpotent ℋ∗-groups is close to that of minimal non- abelian groups. Moreover, if ℋ∞ denotes the class of groups in which all infinite non-abelian subgroups are self-normalizing, it will turn out that any soluble ℋ∞-group either has the property ℋ or is a Černikov group (i.e. if it is a finite extension of an abelian group satisfying the minimal condition on subgroups). Most of our notation is standard and can be found in [7]. 1. Self-normalizing non-abelian subgroups Let G be any ℋ-group. It follows from the definition that G is soluble if and only if its commutator subgroup G′ is properly contained in G; thus every soluble ℋ-group is actually metabelian. Observe also that any group in the class ℋ∞ which properly contains its commutator subgroup either is metabelian or finite-by-abelian. It is well known that any soluble group whose proper subgroups are abelian is either finite or abelian; the structure of finite minimal non- abelian groups is well-known. We will also need the following result of S.N. Černikov [2] concerning groups with few non-abelian subgroups (see also [6]). Lemma 1.1. Let G be a soluble non-abelian group satisfying the minimal condition on non-abelian subgroups. Then G is a Černikov group. Lemma 1.2. Let G be a non-periodic ℋ∞-group. If G′ < G, then G is abelian. Proof. Without loss of generality, we may suppose that G is finitely gener- ated. If G is nilpotent, all its infinite proper subgroups are abelian, so that 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 in particular G satisfies the minimal condition on non-abelian subgroups and hence it is abelian by Lemma 1.1. Assume, to the contrary, that G is not nilpotent, so that G/G′ is a cyclic p-group for some prime number p, and in particular G′ is a finitely generated infinite abelian group. Let x be an element of G such that G = ⟨x,G′⟩, and so G′ = [x,G′]. As ⟨x⟩ ∩G′ ≤ Z(G), the factor group G/⟨x⟩ ∩G′ is infinite by Schur’s the- orem and thus it is also a counterexample. Replacing G by G/⟨x⟩ ∩ G′, we may assume that ⟨x⟩ ∩G′ = {1}. Clearly, G/(G′)p is a finite p-group, so that the proper subgroup ⟨x, (G′)p⟩ is subnormal in G. It follows that ⟨x, (G′)p⟩ is abelian and hence (G′)p = [x,G′]p = [x, (G′)p] = {1}. Therefore G′ is finite and this contradiction completes the proof of the statement. Recall from [1] (p.52) 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. The class of locally graded groups is quite large, and contains in particular all locally (soluble-by-finite) groups. Lemma 1.3. If G is a periodic locally graded ℋ∞-group, then G is locally finite. Proof. Assume for a contradiction that G contains a finitely generated infinite subgroup E. Since G is locally graded, there exists a proper normal subgroup H of E such that E/H is finite. Then H is an infinite proper subgroup of G which is not self-normalizing, so that H is abelian and E is abelian-by-finite. Therefore E is finite, and this contradiction proves that G is locally finite. Our next lemma reduces the study of (soluble) ℋ-groups to the case of non-nilpotent groups. Lemma 1.4. If G is a locally nilpotent ℋ-group, then all proper sub- groups of G are abelian. Proof. Assume for a contradiction that G contains a proper non-abelian subgroup X, and let g be an element of G ∖ X. Clearly, there exists a finitely generated non-abelian subgroup E of X, and E is a proper subnormal subgroup of ⟨E, g⟩, contradicting the property ℋ. Lemma 1.5. If G is a ℋ∗-group, then the factor group G/G′ is cyclic with prime-power order. Jo u rn al A lg eb ra D is cr et e M at h .58 Self-normalizing subgroups Proof. Let x and y be elements of G such that xy ∕= yx. Then ⟨x, y,G′⟩ is a non-abelian normal subgroup of G, so that ⟨x, y,G′⟩ = G and G/G′ is finitely generated. Assume that the statement is false, so that G/G′ contains two distinct maximal subgroups M1/G ′ and M2/G ′. Then the normal subgroups M1 and M2 of G must be abelian, and hence G = M1M2 is nilpotent. It follows from Lemma 1.4 that all proper subgroups of G are abelian, contradicting the hypothesis. The following result is elementary and can be found in [3]. Lemma 1.6. Let A be a periodic abelian group, and let � : A −−→ A be an automorphism of A of prime order p. If A has no elements of order p, then A = CA(�)× [A,�]. We are now in a position to describe all soluble groups in the class ℋ. Theorem 1.7. Let G be a soluble non-nilpotent group. Then G has the property ℋ if and only if G = ⟨x⟩⋉G′, where x is an element of prime- power order pn, xp belongs to CG(G ′) and the commutator subgroup G′ is a periodic abelian group with no elements of order p. Proof. If all proper subgroups of G are abelian, it follows directly from the structure of minimal non-abelian groups that G has the required structure. Suppose now that G is a ℋ∗-group, so that in particular it is periodic by Lemma 1.2. Moreover, Lemma 1.5 yields that there exists a maximal subgroup M of G containing G′, and M is abelian since it is normal in G. Put ∣G : M ∣ = p, and let x be an element of G with order pn (for some n > 0) such that G = ⟨x,M⟩. Write M = A × B, where A is a p′-group and B is a p-group, and assume that the subgroup ⟨x,A⟩ is properly contained in G. Let g be an element of G ∖ ⟨x,A⟩. As ⟨x, g, A⟩/A is a finite p-group, the subgroup ⟨x,A⟩ is properly contained in its normalizer and hence it is abelian. It follows that A is contained in the centre of G. Thus [x,B] ∕= {1} and so the non-abelian p-subgroup ⟨x,B⟩ must be self-normalizing. On the other hand, A is contained in NG(⟨x,B⟩), so that A = {1} and G is a p-group, which is impossible by Lemma 1.4. This contradiction shows that G = ⟨x,A⟩ = ⟨x⟩⋉A. Again Lemma 1.4 yields that the subgroup [A, x] is not contained in Z(G), so that [A, x, x] ∕= {1} and hence ⟨[A, x], x⟩ is not abelian. On the other hand, ⟨[A, x], x⟩ is a normal subgroup of G, so that ⟨[A, x], x⟩ = G and hence A = [A, x] = G′, with xp ∈ M = CG(G ′). Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 59 Conversely, assume that G = ⟨x⟩⋉G′ has the structure described in the statement, and let X be any non-abelian subgroup of G. Obviously, X is not contained in ⟨xp, G′⟩, and so there is a non-trivial element y of X with order a power of p such that ⟨y,G′⟩ = G. Then [G′, y] = G′. Consider the normal subgroup N = G′ ∩ X of G, and put Ḡ = G/N . Then [Ḡ′, ȳ] = Ḡ′ and so also CḠ′(ȳ) = {1} by Lemma 1.6. On the other hand, [y,G′ ∩NG(X)] ≤ G′ ∩X = N, and hence it follows that G′ ∩NG(X) = G′ ∩X. Therefore we have NG(X) = X(G′ ∩NG(X)) = X, ando so G has the property ℋ. 2. Self-normalizing infinite non-abelian subgroups The first lemma of this section extends Lemma 1.4 to the class ℋ∞. Lemma 2.1. Let G be a soluble locally nilpotent ℋ∞-group. Then all infinite proper subgroups of G are abelian. Proof. As the statement is obvious if the group G is nilpotent, we may suppose that G is not nilpotent, so that in particular G′ is infinite and G is not finitely generated. Consider any infinite non-abelian subgroup X of G, and let x and y be elements of X such that xy ∕= yx. Then ⟨x, y,G′⟩ = G and G/G′ is finitely generated. For each element z of G, the non-abelian subgroup ⟨x, y, z⟩ is not self-normalizing and hence it is finite. Therefore G is periodic and G/G′ is finite. Since G′ is abelian, the group G is hypercentral, so that every proper subgroup of G is properly contained in its normalizer; thus all infinite proper subgroups of G are abelian. Lemma 2.2. Let G be an infinite soluble non-abelian group whose infi- nite proper subgroups are abelian. Then G is a Černikov group and its finite residual has no infinite proper G-invariant subgroups. Proof. Clearly G satisfies the minimal condition on non-abelian sub- groups, and hence it is a Černikov group by Lemma 1.1. Let J be the finite residual of G, and let N be any infinite G-invariant subgroup of J . Consider two elements x, y of G such that xy ∕= yx. Then ⟨x, y,N⟩ is an infinite non-abelian subgroup of G, so that ⟨x, y,N⟩ = G and the index ∣G : N ∣ is finite. Therefore N = J and J has no infinite proper G-invariant subgroups. Jo u rn al A lg eb ra D is cr et e M at h .60 Self-normalizing subgroups Lemma 2.3. Let G be an infinite non-abelian ℋ∞-group with finite commutator subgroup. Then G is a finite extension of a group of type p∞ for some prime number p. Proof. The group G is periodic by Lemma 1.2, and so it contains a finite non-abelian subgroup E. Then every proper subgroup of the abelian group G/EG′ must be finite, and hence G/EG′ is a group of type p∞ for some prime number p. As EG′ is finite, it follows that G is a finite extension of a group of type p∞. Lemma 2.4. Let G be a periodic soluble ℋ∞-group with infinite commu- tator subgroup. If N is an infinite G-invariant subgroup of G′ such that G′/N is infinite, then G/G′ is a cyclic p-group for some prime number p and N contains the p-component of G′. Proof. As G′/N is infinite, the factor group Ḡ = G/N neither is abelian nor minimal non-abelian. Moreover, Ḡ has the property ℋ, so that it follows from Theorem 1.7 that Ḡ/Ḡ′ is a cyclic p-group while Ḡ′ is a p′-group for some prime number p. Therefore G/G′ is a cyclic p-group and the p-component of G′ is contained in N . Lemma 2.5. Let G be a soluble p-group with the property ℋ∞. If G′ is infinite, then G is a Černikov group. Proof. Assume first that G is nilpotent. Then all infinite proper sub- groups of G are abelian and hence G is a Černikov group by Lemma 1.1. Suppose now that G is not nilpotent, so that in particular we may con- sider an element x ∈ G ∖ CG(G ′) and G = ⟨x,G′⟩. Thus G′ has finite index in G and so it contains the finite residual J of G. Assume for a contradiction that Ḡ = G/J is infinite. Since Ḡ/Ḡ′ is finite, there exist in Ḡ normal subgroups of finite index H̄ and K̄ such that H̄ < K̄ ≤ Ḡ′ and K̄ ∩ ⟨x̄⟩ = {1}. As Ḡ′ is infinite, the group Ḡ/Z(Ḡ) is likewise infinite and hence the subgroup ⟨x̄, H̄⟩ cannot be abelian; on the other hand, ⟨x̄, H̄⟩ is subnormal in Ḡ since Ḡ/H̄ is a finite p-group. It follows that Ḡ = ⟨x̄, H̄⟩ and so K̄ = ⟨⟨x̄ ∩ K̄⟩, H̄⟩ = H̄. This contradiction shows that G/J is finite, and in particular the abelian subgroup J is divisible. Moreover, Lemma 2.4 yields that the socle of J is finite, so that J satisfies the minimal condition on subgroups and G is a Černikov group. Lemma 2.6. Let G be an infinite non-abelian ℋ∞-group with G′ < G. Then one of the following conditions holds: Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 61 (a) G is a Černikov group and its finite residual has no infinite proper G-invariant subgroups; (b) G = ⟨x⟩⋉G′, where x is an element of prime-power order pn, xp belongs to CG(G ′) and the commutator subgroup G′ is a periodic abelian group with no elements of order p. Proof. It follows from Lemma 1.2 that G is periodic. Moreover, by Lem- ma 2.3 we may assume that G′ is infinite and so abelian, and Lemma 2.2 allows to suppose that G contains some infinite proper non-abelian sub- group, so that in particular G is not nilpotent and hence CG(G ′) is prop- erly contained in G. Let x be an element of prime-power order pn in the set G ∖ CG(G ′). As ⟨x,G′⟩ is an infinite non-abelian normal subgroup of G, we have G = ⟨x,G′⟩, so that G/G′ is a cyclic p-group and G′ = [G′, x]. Write G′ = P ×Q, where P is a p-group and Q is a p′-group. It follows from Lemma 2.4 that one of the subgroups P and Q must be finite. Suppose first that Q is finite, so that the p-subgroup ⟨x, P ⟩ has finite index in G. On the other hand, ⟨x, P ⟩ ≃ G/Q has infinite commutator subgroup and hence it is a Černikov group by Lemma 2.5. Thus G itself is a Černikov group. The finite residual J of G is contained in P and has no proper subgroups of finite index, so that J does not contain infinite proper G-invariant subgroups by Lemma 2.4. Assume now that P is finite. Thus G/Q is a finite p-group and so the infinite subgroup ⟨x,Q⟩ is subnormal in G. Moreover, ⟨x,Q⟩ is not abelian as the commutator subgroup of G is infinite, and then G = ⟨x,Q⟩. It follows that G′ = Q has no elements of order p. Finally, the normal subgroup ⟨xp, G′⟩ is properly contained in G, so that it is abelian and hence C⟨x⟩(G ′) = ⟨xp⟩. The lemma is proved. It follows from Lemma 2.6 that any soluble non-abelian ℋ∞-group either has the property ℋ or is a Černikov group, and hence it allows to restrict our attention to the case of Černikov groups. In fact: Corollary 2.7. Let G be a group such that G′ < G. If G belongs to ℋ∞ ∖ ℋ, then it is a Černikov group. Lemma 2.8. Let G be a Černikov group whose finite residual J has no infinite proper G-invariant subgroups. If the centralizer CG(J) is abelian and G/CG(J) has prime order, then every infinite non-abelian subgroup of G contains J . Proof. Let X be any infinite non-abelian subgroup of G. Then X is not contained in CG(J) and hence G = XCG(J). It follows that the finite residual Y of X is a normal subgroup of G, so that Y = J and J is contained in X. Jo u rn al A lg eb ra D is cr et e M at h .62 Self-normalizing subgroups We also need the following information on finite non-nilpotent groups in the class ℋ. Lemma 2.9. Let G = ⟨x⟩⋉G′ be a finite non-nilpotent ℋ-group, and let A be an abelian subgroup of G. Then either A ≤ CG(G ′) or A = ⟨xg⟩ for some g ∈ G. Proof. Let pn be the order of the element x (where p is a prime); then by Theorem 1.7 the order of G′ is prime to p and CG(G ′) = ⟨xp, G′⟩. Suppose that A is not contained in CG(G ′), so that there exists an element g of G such that xg ∈ A. As CG′(x) = {1}, it follows that A = ⟨xg⟩. Our last two results complete the classification of soluble ℋ∞-groups. Observe that of course a Černikov group is central-by-finite if and only if it has finite commutator subgroup. Theorem 2.10. Let G be a Černikov group such that G′ < G and G/Z(G) is infinite. Then G is a ℋ∞-group if and only if it satisfies one of the following conditions: (a) G has the property ℋ; (b) all infinite proper subgroups of G are abelian; (c) G = ⟨x,G′⟩ for some element x of prime-power order pn, the finite residual J of G is a p-group having no infinite proper G-invariant subgroups, and G′ = J ×K where K is a finite abelian p′-subgroup and C⟨x⟩(G ′) = ⟨xp⟩. Proof. Suppose first that G is a group in the class ℋ∞∖ℋ. It follows from Lemma 2.6 that the finite residual J of G has no infinite proper G-in- variant subgroups; in particular, J is a p-group for some prime number p. Moreover, the centralizer CG(J) is properly contained in G, as G/Z(G) is infinite. Assume that the finite group G/J is nilpotent, and let g be an element of prime-power order qk in the set G ∖ CG(J). Then ⟨g, J⟩ is a non-abelian subnormal subgroup of G, so that ⟨g, J⟩ = G and G/J is cyclic. Clearly, ⟨gq, J⟩ is an infinite proper normal subgroup of G, so that it is abelian and hence ⟨gq, J⟩ = CG(J). Thus G/CG(J) has order q. Let X be any infinite non-abelian subgroup of G. It follows from Lemma 2.8 that X contains J ; on the other hand, X is not contained in CG(J) and hence X = G. Therefore in this case all infinite proper subgroups of G are abelian. Assume now that the finite group G/J is not nilpotent. As G′ is infinite, every proper subgroup of G containing G′ is abelian; thus G/G′ neither is the set-theoretic union of its proper subgroups nor can be gen- erated by two proper subgroups, and hence it is a cyclic q-group for some Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 63 prime number q. In particular, J is contained in G′. As the finite non- nilpotent group G/J has the property ℋ, it follows from Theorem 1.7 that G/J = ⟨xJ⟩ ×G′/J, where x is an element whose order is a power of q and G′/J is a q′-group. The subgroup ⟨xq, G′⟩ must be abelian, and so C⟨x⟩(G ′) = ⟨xq⟩. More- over, [G′, x] = G′ and hence it follows from Lemma 1.6 and Theorem 1.7 that G′ contains elements of order q. Therefore q = p and G′ = J ×K for some finite abelian p′-subgroup K. Conversely, suppose that G satisfies condition (c) of the statement. Then G′ = [G′, x] = [J, x]× [K,x], so that [J, x] = J and [K,x] = K; as C⟨x⟩(K) = ⟨xp⟩, it follows from Lemma 1.6 that CK(x) = {1}. Write C/J = CG′/J(xJ). We have [K ∩ C, x] ≤ J ∩K = {1}, so that K ∩ C = {1} and hence C = JK ∩ C = J × (K ∩ C) = J. Therefore G/J is a finite non-nilpotent ℋ-group. Moreover, as G/Z(G) is infinite, the centralizer CG(J) is properly contained in G, so that CG(J) = ⟨xp, G′⟩; in particular, CG(J) is abelian and G/CG(J) has order p. Let X be any infinite non-abelian subgroup of G. Then X contains J by Lemma 2.8, and so NG(X) if X/J is not abelian. Assume now that X/J is abelian; as X is not contained in ⟨xp, G′⟩, Lemma 2.9 yields that X = ⟨xg, J⟩ for some element g ∈ G. Finally, [K ∩NG(X), X] ≤ K ∩X = {1}, so that K ∩ NG(X) ≤ CK(xg) = {1} and NG(X) = X. Therefore the group G has the property ℋ∞. In the case of central-by-finite groups, the description of the property ℋ∞ must be different. To see this, it is enough to consider a direct product Alt(5)× P, where Alt(5) is the alternating group of degree 5 and P is a group of type p∞ for some prime number p > 5. Theorem 2.11. Let G be an infinite soluble Černikov group such that G/Z(G) is finite. Then G is a ℋ∞-group if and only if it satisfies one of the following conditions: Jo u rn al A lg eb ra D is cr et e M at h .64 Self-normalizing subgroups (a) all infinite proper subgroups of G are abelian; (b) the finite residual J of G is a group of type q∞ for some prime number q, G/J = ⟨xJ⟩⋉G′J/J is a (finite) non-nilpotent ℋ-group and ⟨xp, G′, J⟩ is abelian (where p is the prime number such that the coset xJ has p-power order). Proof. Suppose first that G has the property ℋ∞. If G is nilpotent, all infinite proper subgroups of G are abelian, so that we may assume that G is not nilpotent, and hence the finite group G/J is likewise non-nilpotent since J ≤ Z(G). Of course, G/J is a ℋ-group and so it follows from Theorem 1.7 that G/J = ⟨xJ⟩⋉G′J/J, where the order of the coset xJ is a power of a prime number p. Finally, J is a group of type q∞ for some prime q by Lemma 2.3. Conversely, suppose that G satisfies condition (b) of the statement, and let X be any infinite non-abelian subgroup of G. Then X contains J and X/J cannot be cyclic as J ≤ Z(G). Assume that X/J is abelian, so that it follows from Lemma 2.9 that X/J is contained in CG/J(G ′J/J) = ⟨xp, G′, J⟩/J, contradicting the assumption that ⟨xp, G′, J⟩ is abelian. Thus X/J is not abelian and hence NG(X) = X. Therefore the group G belongs to the class ℋ∞. References [1] B. Amberg, S. Franciosi, F. de Giovanni, Products of Groups, Clarendon Press, 1992. [2] S.N. Černikov, Investigation of groups with given properties of the subgroups, Ukrain. Math. J. 21 (1969), 160–172. [3] S. Franciosi, F. de Giovanni, On torsion groups with nilpotent automorphism groups, Comm. Algebra 14 (1986), 1909–1935. [4] N.F. Kuzennyi, N.N. Semko, Structure of solvable nonnilpotent metahamiltonian groups, Math. Notes 34 (1983), 572–577. [5] V.N. Obraztsov, An embedding theorem for groups and its corollaries, USSR Sb. 66 (1990), 541–553. [6] R.E. Phillips, J.S. Wilson, On certain minimal conditions for infinite groups, J. Algebra 51 (1978), 41–68. [7] D.J.S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, 1972. [8] G.M. Romalis, N.F. Sesekin, Metahamiltonian groups, Ural. Gos. Univ. Mat. Zap. 5 (1966), 101–106. Jo u rn al A lg eb ra D is cr et e M at h .M. De Falco, F. de Giovanni, C. Musella 65 [9] G.M. Romalis, N.F. Sesekin, Metahamiltonian groups II, Ural. Gos. Univ. Mat. Zap. 6 (1968), 52–58. [10] G.M. Romalis, N.F. Sesekin, Metahamiltonian groups III, Ural. Gos. Univ. Mat. Zap. 7 (1969/70), 195–199.g 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: 09.08.2009 and in final form 09.08.2009.

Groups with many self-normalizing subgroups

Institution

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