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Journal articles on the topic 'Soluble and nilpotent group'

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Published: 10 March 2023

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1

Burns, R. G., and Yuri Medvedev. "Group Laws Implying Virtual Nilpotence." Journal of the Australian Mathematical Society 74, no. 3 (June 2003): 295–312. http://dx.doi.org/10.1017/s1446788700003335.

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AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

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2

Amberg, Bernhard, Silvana Franciosi, and Francesco de Giovanni. "Nilpotent-by-Noetherian Factorized Groups." Canadian Mathematical Bulletin 32, no. 4 (December 1, 1989): 391–403. http://dx.doi.org/10.4153/cmb-1989-057-8.

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AbstractIt is shown that a soluble-by-finite product G = AB of a nilpotent-by-noetherian group A and a noetherian group B is nilpotentby- noetherian. Moreover, a bound for the torsion-free rank of the Fitting factor group of G is given, in terms of the torsion-free rank of the Fitting factor group of A and the torsion-free rank of B.

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3

HU, BIN, JIANHONG HUANG, and ALEXANDER N. SKIBA. "ON THE σ-NILPOTENT NORM AND THE σ-NILPOTENT LENGTH OF A FINITE GROUP." Glasgow Mathematical Journal 63, no. 1 (February 27, 2020): 121–32. http://dx.doi.org/10.1017/s0017089520000051.

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AbstractLet G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes $\Bbb{P}$ . Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use $G^{{\mathfrak{N}}_{\sigma}}$ to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is, $${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$ Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.

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4

Shi, Jiangtao, and Cui Zhang. "A Note on TI-Subgroups of a Finite Group." Algebra Colloquium 21, no. 02 (April 11, 2014): 343–46. http://dx.doi.org/10.1142/s1005386714000297.

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Let G be a finite group and H a subgroup of G. Recall that H is said to be a TI-subgroup of G if Hg∩ H = 1 or H for each g ∈ G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group G are TI-subgroups, then G is soluble, and all non-nilpotent subgroups of G are normal.

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5

Robinson, Derek J. S. "Soluble products of nilpotent groups." Journal of Algebra 98, no. 1 (January 1986): 183–96. http://dx.doi.org/10.1016/0021-8693(86)90021-9.

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6

Smith, Howard. "Groups with few non-nilpotent subgroups." Glasgow Mathematical Journal 39, no. 2 (May 1997): 141–51. http://dx.doi.org/10.1017/s0017089500032031.

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Let G be a non-nilpotent group in which all proper subgroups are nilpotent. If G is finite then G is soluble [18], and a classification of such groups is given in [14]. The paper [12]. of Newman and Wiegold discusses infinite groups with this property. Clearly such a group is either finitely generated or locally nilpotent. Many interesting results concerning the finitely generated case are established in [12]. Since the publication of that paper there have appeared the examples due to Ol'shanskii and Rips (see [13]) of finitely generated infinite simple p-groups all of whose proper nontrivial subgroups have order p, a prime. Following [12], let us say that a group G is an AN-group if it is locally nilpotent and non-nilpotent with all proper subgroups nilpotent. A complete description is given in Section 4 of [12] of AN-groups having maximal subgroups. Every soluble AN-gvoup has locally cyclic derived factor group and is a p-group for some prime p ([12; Lemma 4.2]). The only further information provided in [12] on AN-groups without maximal subgroups is that they are countable. Four years or so after the publication of [12], there appeared the examples of Heineken and Mohamed [5]: for every prime p there exists a metabelian, non-nilpotent p-group G having all proper subgroups nilpotent and subnormal; further, G has no maximal subgroups and so G/G' is a Prüfer p-group in each case.

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7

Milliet, Cédric. "On the definability of radicals in supersimple groups." Journal of Symbolic Logic 78, no. 2 (June 2013): 649–56. http://dx.doi.org/10.2178/jsl.7802160.

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AbstractIf G is a group with a supersimple theory having a finite SU-rank, then the subgroup of G generated by all of its normal nilpotent subgroups is definable and nilpotent. This answers a question asked by Elwes, Jaligot, Macpherson and Ryten. If H is any group with a supersimple theory, then the subgroup of H generated by all of its normal soluble subgroups is definable and soluble.

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8

Franciosi, Silvana, Francesco de Giovanni, and Yaroslav P. Sysak. "An extension of the Kegel–Wielandt theorem to locally finite groups." Glasgow Mathematical Journal 38, no. 2 (May 1996): 171–76. http://dx.doi.org/10.1017/s0017089500031402.

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A famous theorem of Kegel and Wielandt states that every finite group which is the product of two nilpotent subgroups is soluble (see [1], Theorem 2.4.3). On the other hand, it is an open question whether an arbitrary group factorized by two nilpotent subgroups satisfies some solubility condition, and only a few partial results are known on this subject. In particular, Kegel [6] obtained an affirmative answer in the case of linear groups, and in the same article he also proved that every locally finite group which is the product of two locally nilpotent FC-subgroups is locally soluble. Recall that a group G is said to be an FC-group if every element of G has only finitely many conjugates. Moreover, Kazarin [5] showed that if the locally finite group G = AB is factorized by an abelian subgroup A and a locally nilpotent subgroup B, then G is locally soluble. The aim of this article is to prove the following extension of the Kegel–Wielandt theorem to locally finite products of hypercentral groups.

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9

Franciosi, Silvana, and Francesco de Giovanni. "On the Hirsch-Plotkin radical of a factorized group." Glasgow Mathematical Journal 34, no. 2 (May 1992): 193–99. http://dx.doi.org/10.1017/s0017089500008715.

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Let the group G = AB be the product of two subgroups A and B. A normal subgroup K of G is said to be factorized if K = (A ∩ K)(B ∩ K) and A ∩ B ≤ K, and this is well-known to be equivalent to the fact that K = AK ∩ BK (see [1]). Easy examples show that normal subgroups of a product of two groups need not, in general, be factorized. Therefore the determination of certain special factorized subgroups is of relevant interest in the investigation concerning the structure of a factorized group. In this direction E. Pennington [5] proved that the Fitting subgroup of a finite product of two nilpotent groups is factorized. This result was extended to infinite groups by B. Amberg and theauthors, who provedin [2] that if the soluble group G = AB with finite abelian section rank isthe product of two locally nilpotent subgroups A and B, then the Hirsch-Plotkin radical (i.e. the maximum locally nilpotent normal subgroup) of G is factorized. If G is a soluble ℒI group and the factors A and B are nilpotent, it was shown in [3] that also the Fitting subgroup of G is factorized. However, Pennington's theorem becomes false for finite soluble groups which are the productof two arbitrary subgroups. For instance, the symmetric group of degree 4 is the product of a subgroup isomorphic with the symmetric group of degree 3 and a cyclic subgroup of order 4, but its Fitting subgroup is not factorized.

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10

Taeri, Bijan. "A question of Paul Erdös and nilpotent-by-finite groups." Bulletin of the Australian Mathematical Society 64, no. 2 (October 2001): 245–54. http://dx.doi.org/10.1017/s0004972700039903.

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Let n be a positive integer or infinity (denoted ∞), k a positive integer. We denote by Ωk(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist distinct elements x, y ∈ X and integers t0, t1…, tk such that , where xi, ∈ {x, y}, i = 0, 1,…,k, x0 ≠ x1. If the integers t0, t1,…,tk are the same for any subset X of G, we say that G is in the class Ω̅k(n). The class k (n) is defined exactly as Ωk(n) with the additional conditions . Let t2, t3,…,tk be fixed integers. We denote by the class of all groups G such that for any infinite subsets X and Y there exist x ∈ X, y ∈ Y such that , where xi ∈ {x, y}, x0 ≠ x1, i = 2, 3, …, k. Here we prove that (1) If G ∈ k(2) is a finitely generated soluble group, then G is nilpotent.(2) If G ∈ Ωk(∞) is a finitely generated soluble group, then G is nilpotentby-finite.(3) If G ∈ Ω̅k(n), n a positive integer, is a finitely generated residually finite group, then G is nilpotent-by-finite.(4) If G is an infinite -group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient, then G is nilpotent-by-finite.

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11

ENDIMIONI, GÉRARD, and GUNNAR TRAUSTASON. "ON VARIETIES IN WHICH SOLUBLE GROUPS ARE TORSION-BY-NILPOTENT." International Journal of Algebra and Computation 15, no. 03 (June 2005): 537–45. http://dx.doi.org/10.1142/s0218196705002402.

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12

Leinen, Felix. "Uncountable existentially closed groups in locally finite group classes." Glasgow Mathematical Journal 32, no. 2 (May 1990): 153–63. http://dx.doi.org/10.1017/s0017089500009174.

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In this paper, will always denote a local class of locally finite groups, which is closed with respect to subgroups, homomorphic images, extensions, and with respect to cartesian powers of finite -groups. Examples for x are the classes L ℐπ of all locally finite π-groups and L(ℐπ ∩ ) of all locally soluble π-groups (where π is a fixed set of primes). In [4], a wreath product construction was used in the study of existentially closed -groups (=e.c. -groups); the restrictive type of construction available in [4] permitted results for only countable groups. This drawback was then removed partially in [5] with the help of permutational products. Nevertheless, the techniques essentially only permitted amalgamation of -groups with locally nilpotent π-groups. Thus, satisfactory results could be obtained for Lp-groups (resp. locally nilpotent π-groups) [6], while the theory remained incomplete in all other cases.

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13

MILLIET, CÉDRIC. "ON THE RADICALS OF A GROUP THAT DOES NOT HAVE THE INDEPENDENCE PROPERTY." Journal of Symbolic Logic 81, no. 4 (August 12, 2016): 1444–50. http://dx.doi.org/10.1017/jsl.2015.56.

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AbstractWe give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable and whose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013.

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14

Heineken, Hermann. "Nilpotent subgroups of finite soluble groups." Archiv der Mathematik 56, no. 5 (May 1991): 417–23. http://dx.doi.org/10.1007/bf01200083.

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15

Ballester-Bolinches, A., M. D. Pérez-Ramos, and A. Martínez- Pastor. "Nilpotent-like fitting formations of finite soluble groups." Bulletin of the Australian Mathematical Society 62, no. 3 (December 2000): 427–33. http://dx.doi.org/10.1017/s0004972700018943.

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Dedicated to Professor K. Doerk on his 60th Birthday.In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.

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16

ENDIMIONI, GÉRARD. "CHARACTERISTIC RELATIONS FOR A FINITE-BY-NILPOTENT GROUP." International Journal of Algebra and Computation 15, no. 02 (April 2005): 273–77. http://dx.doi.org/10.1142/s0218196705002219.

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We improve previous results by showing that a finitely generated soluble group G is finite-by-nilpotent if and only if for all a, b ∈ G, there exists a positive integer n such that [a, nb] belongs to γn+2(<a, b>).<a<a,b>>′.

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17

Smith, Howard, and James Wiegold. "Soluble groups isomorphic to their non-nilpotent subgroups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 67, no. 3 (December 1999): 399–411. http://dx.doi.org/10.1017/s1446788700002081.

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AbstractA group G belongs to the class W if G has non-nilpotent proper subgroups and is isomorphic to all of them. The main objects of study are the soluble groups in W that are not finitely generated. It is proved that there are no torsion-free groups of this sort, and a reasonable classification is given in the finite rank case.

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18

Jaber, Khaled. "Équations génériques dans un groupe stable nilpotent." Journal of Symbolic Logic 64, no. 2 (June 1999): 761–68. http://dx.doi.org/10.2307/2586498.

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AbstractWe prove that in a nilpotent-by-finite stable group an equation that holds generically holds everywhere. Combining this result with results of Wagner and Bryant, we conclude that a soluble-by-finite stable group of generic exponent n has exponent n.

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19

Woods, William. "On the structure of virtually nilpotent compact p-adic analytic groups." Journal of Group Theory 21, no. 1 (January 1, 2018): 165–88. http://dx.doi.org/10.1515/jgth-2017-0017.

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AbstractLetGbe a compactp-adic analytic group. We recall the well-understood finite radical{\Delta^{+}}and FC-centre Δ, and introduce ap-adic analogue of Roseblade’s subgroup{\mathrm{nio}(G)}, the unique largest orbitally sound open normal subgroup ofG. Further, whenGis nilpotent-by-finite, we introduce the finite-by-(nilpotentp-valuable) radical{\mathbf{FN}_{p}(G)}, an open characteristic subgroup ofGcontained in{\mathrm{nio}(G)}. By relating the already well-known theory of isolators with Lazard’s notion ofp-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble)p-valuable group, and use this to study the conjugation action of{\mathrm{nio}(G)}on{\mathbf{FN}_{p}(G)}. We emerge with a structure theorem forG,1\leq\Delta^{+}\leq\Delta\leq\mathbf{FN}_{p}(G)\leq\mathrm{nio}(G)\leq G,in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group ringskG) of such groups, and will be used in future work to study the prime ideals of these rings.

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20

SU, NING, and YANMING WANG. "ON THE -LENGTH AND THE WIELANDT LENGTH OF A FINITE -SOLUBLE GROUP." Bulletin of the Australian Mathematical Society 88, no. 3 (March 7, 2013): 453–59. http://dx.doi.org/10.1017/s0004972713000026.

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AbstractThe $p$-length of a finite $p$-soluble group is an important invariant parameter. The well-known Hall–Higman $p$-length theorem states that the $p$-length of a $p$-soluble group is bounded above by the nilpotent class of its Sylow $p$-subgroups. In this paper, we improve this result by giving a better estimation on the $p$-length of a $p$-soluble group in terms of other invariant parameters of its Sylow $p$-subgroups.

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21

ENDIMIONI, GÉRARD. "NORMAL AUTOMORPHISMS OF A FREE METABELIAN NILPOTENT GROUP." Glasgow Mathematical Journal 52, no. 1 (December 4, 2009): 169–77. http://dx.doi.org/10.1017/s0017089509990267.

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AbstractAn automorphism φ of a group G is said to be normal if φ(H) = H for each normal subgroup H of G. These automorphisms form a group containing the group of inner automorphisms. When G is a non-abelian free (or free soluble) group, it is known that these groups of automorphisms coincide, but this is not always true when G is a free metabelian nilpotent group. The aim of this paper is to determine the group of normal automorphisms in this last case.

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22

Ballester-Bolinches, A., A. Martínez-Pastor, and M. D. Pérez-Ramos. "A family of dominant Fitting classes of finite soluble groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 64, no. 1 (February 1998): 33–43. http://dx.doi.org/10.1017/s1446788700001270.

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AbstractIn this paper a large family of dominant Fitting classes of finite soluble groups and the description of the corresponding injectors are obtained. Classical constructions of nilpotent and Lockett injectors as well as p-nilpotent injectors arise as particular cases.

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23

Endimioni, Gérard. "Groups covered by finitely many nilpotent subgroups." Bulletin of the Australian Mathematical Society 50, no. 3 (December 1994): 459–64. http://dx.doi.org/10.1017/s0004972700013575.

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Let G be a finitely generated soluble group. Lennox and Wiegold have proved that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that (x, y) is nilpotent. The main theorem of this paper is an improvement of the previous result: we show that G has a finite covering by nilpotent subgroups if and only if any infinite set of elements of G contains a pair {x, y} such that [x, ny] = 1 for some integer n = n(x, y) ≥ 0.

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24

Asiáin, M. J. "Classes of generalized nilpotent and soluble groups." Siberian Mathematical Journal 37, no. 3 (May 1996): 423–29. http://dx.doi.org/10.1007/bf02104843.

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25

Delizia, Costantino, Primoz Moravec, and Chiara Nicotera. "Finite groups in which some property of two-generator subgroups is transitive." Bulletin of the Australian Mathematical Society 75, no. 2 (April 2007): 313–20. http://dx.doi.org/10.1017/s000497270003923x.

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Finite groups in which a given property of two-generator subgroups is a transitive relation are investigated. We obtain a description of such groups and prove in particular that every finite soluble-transitive group is soluble. A classification of finite nilpotent-transitive groups is also obtained.

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26

DE GIOVANNI, FRANCESCO, and MARCO TROMBETTI. "NILPOTENCY IN UNCOUNTABLE GROUPS." Journal of the Australian Mathematical Society 103, no. 1 (October 27, 2016): 59–69. http://dx.doi.org/10.1017/s1446788716000379.

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The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.

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27

Heliel, Abd El-Rahman, Mohammed Al-Shomrani, and Adolfo Ballester-Bolinches. "On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups." Mathematics 8, no. 12 (December 4, 2020): 2165. http://dx.doi.org/10.3390/math8122165.

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Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.

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28

Trabelsi, Nadir. "Characterisation of nilpotent-by-finite groups." Bulletin of the Australian Mathematical Society 61, no. 1 (February 2000): 33–38. http://dx.doi.org/10.1017/s0004972700021997.

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LetGbe a finitely generated soluble group. The main result of this note is to prove thatGis nilpotent-by-finite if, and only if, for every pairX,Yof infinite subsets ofG, there exist anxinX,yinYand two positive integersm=m(x,y),n=n(x,y) satisfying [x,nym] = 1. We prove also that ifGis infinite and ifmis a positive integer, thenGis nilpotent-by-(finite of exponent dividingm) if, and only if, for every pairX,Yof infinite subsets ofG, there exist anxinX,yinYand a positive integern=n(x,y) satisfying [x,nym] = 1.

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JAFARIAN AMIRI, S. M., H. MADADI, and H. ROSTAMI. "ON THE PROBABILITY OF GENERATING NILPOTENT SUBGROUPS IN A FINITE GROUP." Bulletin of the Australian Mathematical Society 93, no. 3 (November 20, 2015): 447–53. http://dx.doi.org/10.1017/s0004972715001252.

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Let $G$ be a finite group. We denote by ${\it\nu}(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup and by $\text{Nil}_{G}(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup. A group $G$ is called an ${\mathcal{N}}$-group if $\text{Nil}_{G}(x)$ is a subgroup of $G$ for all $x\in G$. We prove that if $G$ is an ${\mathcal{N}}$-group with ${\it\nu}(G)>\frac{1}{12}$, then $G$ is soluble. Also, we classify semisimple ${\mathcal{N}}$-groups with ${\it\nu}(G)=\frac{1}{12}$.

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30

Lv, Heng, Chong Zhao, and Wei Zhou. "Soluble groups with few conjugate classes of non-cyclic subgroups." Journal of Algebra and Its Applications 18, no. 06 (May 27, 2019): 1950114. http://dx.doi.org/10.1142/s0219498819501147.

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For a group [Formula: see text], let [Formula: see text] be the number of conjugate classes of the non-cyclic subgroups. In this paper, we prove that the derived length of the group [Formula: see text] with [Formula: see text] is at most 3, and we also study the non-nilpotent group [Formula: see text] with [Formula: see text].

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31

Abdollahi, Alireza. "Some Engel conditions on infinite subsets of certain groups." Bulletin of the Australian Mathematical Society 62, no. 1 (August 2000): 141–48. http://dx.doi.org/10.1017/s0004972700018554.

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Let k be a positive integer. We denote by ɛk(∞) the class of all groups in which every infinite subset contains two distinct elements x, y such that [x,k y] = 1. We say that a group G is an -group provided that whenever X, Y are infinite subsets of G, there exists x ∈ X, y ∈ Y such that [x,k y] = 1. Here we prove that:(1) If G is a finitely generated soluble group, then G ∈ ɛ3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3.(2) If G is a finitely generated metabelian group, then G ∈ ɛk(∞) if and only if G/Zk (G) is finite, where Zk (G) is the (k + 1)-th term of the upper central series of G.(3) If G is a finitely generated soluble ɛk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt (G) is finite.(4) If G is an infinite -group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.

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32

Burns, R. G., and Yuri Medvedev. "A note on Engel groups and local nilpotence." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 64, no. 1 (February 1998): 92–100. http://dx.doi.org/10.1017/s1446788700001324.

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AbstractThis paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

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33

Monakhov, Victor, and Irina Sokhor. "On cofactors of subnormal subgroups." Journal of Algebra and Its Applications 15, no. 09 (August 22, 2016): 1650169. http://dx.doi.org/10.1142/s0219498816501693.

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For a soluble finite group [Formula: see text] and a prime [Formula: see text] we let [Formula: see text], [Formula: see text]. We obtain upper bounds for the rank, the nilpotent length, the derived length, and the [Formula: see text]-length of a finite soluble group [Formula: see text] in terms of [Formula: see text] and [Formula: see text].

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34

du Sautoy, Marcus. "ZETA FUNCTIONS OF GROUPS: EULER PRODUCTS AND SOLUBLE GROUPS." Proceedings of the Edinburgh Mathematical Society 45, no. 1 (February 2002): 149–54. http://dx.doi.org/10.1017/s0013091500000456.

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AbstractThe well-behaved Sylow theory for soluble groups is exploited to prove an Euler product for zeta functions counting certain subgroups in pro-soluble groups. This generalizes a result of Grunewald, Segal and Smith for nilpotent groups.AMS 2000 Mathematics subject classification: Primary 20F16; 11M99

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SHUMYATSKY, PAVEL, and CARMELA SICA. "ON THE STRUCTURE OF SOLUBLE GRADED LIE ALGEBRAS." Journal of Algebra and Its Applications 10, no. 04 (August 2011): 597–604. http://dx.doi.org/10.1142/s021949881100477x.

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Let A be the elementary group of order 2n and L an A-graded Lie algebra with L0 = 0. Assume that L is soluble with derived length k. It is proved that L has a series of ideals of length n all of whose quotients are nilpotent of {k, n}-bounded class.

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36

Lu, Jiakuan. "On a theorem of Gagola and Lewis." Journal of Algebra and Its Applications 16, no. 08 (August 17, 2016): 1750158. http://dx.doi.org/10.1142/s0219498817501584.

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Gagola and Lewis proved that a finite group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible characters [Formula: see text] of [Formula: see text]. In this paper, we prove that a finite soluble group [Formula: see text] is nilpotent if and only if [Formula: see text] divides [Formula: see text] [Formula: see text] [Formula: see text] for all irreducible monomial characters [Formula: see text] of [Formula: see text].

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37

Smith, Howard, and James Wiegold. "Groups isomorphic to their non-nilpotent subgroups." Glasgow Mathematical Journal 40, no. 2 (May 1998): 257–62. http://dx.doi.org/10.1017/s0017089500032572.

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We were concerned in [12] with groups G that are isomorphic to all of their non-abelian subgroups. In order to exclude groups with all proper subgroups abelian, which are well understood in the finite case [7] and which include Tarski groups in the infinite case, we restricted attention to the class X of groups G that are isomorphic to their nonabelian subgroups and that contain proper subgroups of this type; such groups are easily seen to be 2-generator, and a complete classification was given in [12, Theorem 2] for the case G soluble. In the insoluble case, G/Z(G) is infinite simple [12; Theorem 1], though not much else was said in [12] about such groups. Here we examine a property which represents a natural generalisation of that discussed above. Let us say that a group G belongs to the class W if G is isomorphic to each of its non-nilpotent subgroups and not every proper subgroup of G is nilpotent. Firstly, note that finite groups in which all proper subgroups are nilpotent are (again) well understood [9]. In addition, much is known about infinite groups with all proper subgroups nilpotent (see, in particular, [8] and [13] for further discussion) although, even in the locally nilpotent case, there are still some gaps in our understanding of such groups. We content ourselvesin the present paper with discussing finitely generated W-groups— note that a W-group is certainly finitely generated or locally nilpotent. We shall have a little more to say about the locally nilpotent case below.

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GROVES, DANIEL. "SOME PROPERTIES OF FREE GROUPS OF SOME SOLUBLE VARIETIES OF GROUPS." Journal of the London Mathematical Society 63, no. 3 (June 2001): 592–606. http://dx.doi.org/10.1017/s0024610701002022.

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Let F be a free group, and let γn(F) be the nth term of the lower central series of F. It is proved that F/[γj(F), γi(F), γk(F)] and F/[γj(F), γi(F), γk(F), γl(F)] are torsion free and residually nilpotent for certain values of i, j, k and i, j, k, l, respectively. In the process of proving this, it is proved that the analogous Lie rings are torsion free.

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Rhemtulla, A. H., and J. S. Wilson. "On Elliptically Embedded Subgroups of Soluble Groups." Canadian Journal of Mathematics 39, no. 4 (August 1, 1987): 956–68. http://dx.doi.org/10.4153/cjm-1987-048-6.

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We call a subset X of a group an elliptic set if there is an integer n such that each element of the group generated by X can be written as a product of at most n elements of X ∪ X−1. The terminology is due to Philip Hall, who investigated elliptic sets in lectures given in Cambridge in the 1960's. Hall was chiefly interested in sets X which are unions of conjugacy classes, but among other things he proved that if H, K are subgroups of a finitely generated nilpotent group then their union H ∪ K is elliptic. We shall say that a subgroup H of an arbitrary group G is elliptically embedded in G, and we write H ee G, if H ∪ K is an elliptic set for each subgroup K of G. Thus H ee G if for each subgroup K there is an integer n (depending on K) such thatwhere the product has 2n factors.

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40

Asiáin, M. J. "Nilpotent and soluble groups with respect to a homomorph." Communications in Algebra 22, no. 6 (January 1994): 1945–54. http://dx.doi.org/10.1080/00927879408824949.

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Lennox, John C., Peter M. Neumann, and James Wiegold. "Nilpotent subgroups and the hypercentre of infinite soluble groups." Archiv der Mathematik 54, no. 5 (May 1990): 417–21. http://dx.doi.org/10.1007/bf01188666.

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42

Alizadeh Sanati, Mahboubeh. "Commutator Length of Finitely Generated Linear Groups." International Journal of Mathematics and Mathematical Sciences 2008 (2008): 1–5. http://dx.doi.org/10.1155/2008/281734.

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The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

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Khosravi, H., and A. Faramarzi Salles. "Groups which satisfy a Thue–Morse identity." Journal of Algebra and Its Applications 19, no. 10 (October 18, 2019): 2050191. http://dx.doi.org/10.1142/s0219498820501911.

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In this paper, we study 3-Thue–Morse groups, but these are the groups satisfying the semigroup identity [Formula: see text]. We prove that if [Formula: see text] is a 3-Thue–Morse group then [Formula: see text] is soluble for every [Formula: see text] and [Formula: see text] in [Formula: see text]. Furthermore, if [Formula: see text] is an Engel group without involution then we show that [Formula: see text] is locally nilpotent.

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CHI, ZHANG, and ALEXANDER N. SKIBA. "ON A LATTICE CHARACTERISATION OF FINITE SOLUBLE PST-GROUPS." Bulletin of the Australian Mathematical Society 101, no. 2 (July 10, 2019): 247–54. http://dx.doi.org/10.1017/s0004972719000741.

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Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

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Trabelsi, Nadir. "Soluble groups with many 2-generator torsion-by-nilpotent subgroups." Publicationes Mathematicae Debrecen 67, no. 1-2 (July 1, 2005): 93–102. http://dx.doi.org/10.5486/pmd.2005.3061.

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46

Heinenken, Hermann. "Fitting classes of certain metanilpotent groups." Glasgow Mathematical Journal 36, no. 2 (May 1994): 185–95. http://dx.doi.org/10.1017/s001708950003072x.

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There are two families of group classes that are of particular interest for clearing up the structure of finite soluble groups: Saturated formations and Fitting classes. In both cases there is a unique conjugacy class of subgroups which are maximal as members of the respective class combined with the property of being suitably mapped by homomorphisms (in the case of saturated formations) or intersecting suitably with normal subgroups (when considering Fitting classes). While it does not seem too difficult, however, to determine the smallest saturated formation containing a given group, the same problem regarding Fitting classes does not seem answered for the dihedral group of order 6. The object of this paper is to determine the smallest Fitting class containing one of the groups described explicitly later on; all of them are qp-groups with cyclic commutator quotient group and only one minimal normal subgroup which in addition coincides with the centre. Unlike the results of McCann [7], which give a determination “up to metanilpotent groups”, the description is complete in this case. Another family of Fitting classes generated by a metanilpotent group was considered and described completely by Hawkes (see [5, Theorem 5.5 p. 476]); it was shown later by Brison [1, Proposition 8.7, Corollary 8.8], that these classes are in fact generated by one finite group. The Fitting classes considered here are not contained in the Fitting class of all nilpotent groups but every proper Fitting subclass is. They have the following additional properties: all minimal normal subgroups are contained in the centre (this follows in fact from Gaschiitz [4, Theorem 10, p. 64]) and the nilpotent residual is nilpotent of class two (answering the open question on p. 482 of Hawkes [5]), while the quotient group modulo the Fitting subgroup may be nilpotent of any class. In particular no one of these classes consists of supersoluble groups only.

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Endimioni, Gérard, and Carmela Sica. "Centralizer of Engel Elements in a Group." Algebra Colloquium 17, no. 03 (September 2010): 487–94. http://dx.doi.org/10.1142/s1005386710000465.

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In this paper we show that some finiteness properties on a centralizer of a particular subgroup can be inherited by the whole group. Among other things, we prove the following characterization of polycyclic groups: a soluble group G is polycyclic if and only if it contains a finitely generated subgroup H, formed by bounded left Engel elements, whose centralizer CG(H) is polycyclic. In the context of Černikov groups we obtain a more general result: a radical group is a Černikov group if and only if it contains a finitely generated subgroup, formed by left Engel elements, whose centralizer is a Černikov group. The aforementioned results generalize a theorem by Onishchuk and Zaĭtsev about the centralizer of a finitely generated subgroup in a nilpotent group.

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KAZARIN, L. S., A. MARTÍNEZ-PASTOR, and M. D. PÉREZ-RAMOS. "ON SYLOW NORMALIZERS OF FINITE GROUPS." Journal of Algebra and Its Applications 13, no. 03 (October 31, 2013): 1350116. http://dx.doi.org/10.1142/s0219498813501168.

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The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

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Cossey, John, and Yanming Wang. "Finite dinilpotent groups of small derived length." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 67, no. 3 (December 1999): 318–28. http://dx.doi.org/10.1017/s1446788700002044.

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AbstractA finite dinilpotent group G is one that can be written as the product of two finite nilpotent groups, A and B say. A finite dinilpotent group is always soluble. If A is abelian and B is metabelian, with |A| and|B| coprime, we show that a bound on the derived length given by Kazarin can be improved. We show that G has derived length at most 3 unless G contains a section with a well defined structure: in particular if G is of odd order, G has derived length at most 3.

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50

Baumeister, Barbara, and Gil Kaplan. "Critical maximal subgroups and conjugacy of supplements in finite soluble groups." Journal of Group Theory 21, no. 1 (January 1, 2018): 45–63. http://dx.doi.org/10.1515/jgth-2017-0033.

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AbstractLetGbe a finite group with an abelian normal subgroupN. When doesNhave a unique conjugacy class of complements inG? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups ofGclosed under conjugation whose intersection equals{\Phi(G)}. In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when{\Phi(G)=1}, these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.

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