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Journal articles on the topic 'Locally nilpotent'

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Author: Grafiati

Published: 10 March 2023

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1

HAVAS, GEORGE, and M. R. VAUGHAN-LEE. "4-ENGEL GROUPS ARE LOCALLY NILPOTENT." International Journal of Algebra and Computation 15, no. 04 (August 2005): 649–82. http://dx.doi.org/10.1142/s0218196705002475.

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Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

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2

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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3

Wehrfritz, B. A. F. "Some nilpotent and locally nilpotent matrix groups." Journal of Pure and Applied Algebra 60, no. 3 (October 1989): 289–312. http://dx.doi.org/10.1016/0022-4049(89)90089-3.

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4

TRAUSTASON, GUNNAR. "A NOTE ON THE LOCAL NILPOTENCE OF 4-ENGEL GROUPS." International Journal of Algebra and Computation 15, no. 04 (August 2005): 757–64. http://dx.doi.org/10.1142/s021819670500244x.

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Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth–Bendix algorithm. In this paper we give a short handproof of the nilpotency of T.

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5

Liao, Jun, and Yanjun Liu. "Minimal Non-nilpotent and Locally Nilpotent Fusion Systems." Algebra Colloquium 23, no. 03 (June 20, 2016): 455–62. http://dx.doi.org/10.1142/s1005386716000432.

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The main purpose of this note is to show that there is a one-to-one correspondence between minimal non-nilpotent (resp., locally nilpotent) saturated fusion systems and finite p′-core-free p-constrained minimal non-nilpotent (resp., locally p-nilpotent) groups.

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6

Detinko, A. S., and D. L. Flannery. "Locally Nilpotent Linear Groups." Irish Mathematical Society Bulletin 0056 (2005): 37–51. http://dx.doi.org/10.33232/bims.0056.37.51.

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7

Karaś, Marek. "Locally Nilpotent Monomial Derivations." Bulletin of the Polish Academy of Sciences Mathematics 52, no. 2 (2004): 119–21. http://dx.doi.org/10.4064/ba52-2-2.

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8

SHUMYATSKY, PAVEL. "A (locally nilpotent)-by-nilpotent variety of groups." Mathematical Proceedings of the Cambridge Philosophical Society 132, no. 2 (March 2002): 193–96. http://dx.doi.org/10.1017/s0305004102005571.

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9

Longobardi, Patrizia, Mercede Maj, Howard Smith, and James Wiegold. "Torsion-free groups isomorphic to all of their non-nilpotent subgroups." Journal of the Australian Mathematical Society 71, no. 3 (December 2001): 339–48. http://dx.doi.org/10.1017/s1446788700002974.

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AbstractThe main result is that every torsion-free locally nilpotent group that is isomorphic to each of its nonnilpotent subgroups is nilpotent, that is, a torsion-free locally nilpotent group G that is not nilpotent has a non-nilpotent subgroup H that is not isomorphic to G.

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10

TRAUSTASON, GUNNAR. "TWO GENERATOR 4-ENGEL GROUPS." International Journal of Algebra and Computation 15, no. 02 (April 2005): 309–16. http://dx.doi.org/10.1142/s0218196705002189.

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Using known results on 4-Engel groups one can see that a 4-Engel group is locally nilpotent if and only if all its 3-generator subgroups are nilpotent. As a step towards settling the question whether all 4-Engel groups are locally nilpotent we show that all 2-generator 4-Engel groups are nilpotent.

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11

Amberg, Bernhard, Silvana Franciosi, and Francesco de Giovanni. "Groups with a nilpotent triple factorisation." Bulletin of the Australian Mathematical Society 37, no. 1 (February 1988): 69–79. http://dx.doi.org/10.1017/s0004972700004159.

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In the investigation of factorised groups one often encounters groups G = AB = AK = BK which have a triple factorisation as a product of two subgroups A and B and a normal subgroup K of G. It is of particular interest to know whether G satisfies some nilpotency requirement whenever the three subgroups A, B and K satisfy this same nilpotency requirement. A positive answer to this problem for the classes of nilpotent, hypercentral and locally nilpotent groups is given under the hypothesis that K is a minimax group or G has finite abelian section rank. The results become false if K has only finite Prüfer rank. Some applications of the main theorems are pointed out.

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12

Kelarev, A. V., and J. Okniński. "On group graded rings satisfying polynomial identities." Glasgow Mathematical Journal 37, no. 2 (May 1995): 205–10. http://dx.doi.org/10.1017/s0017089500031104.

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A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18]). Several interesting results of this sort have appeared in the literature recently. In particular, it was proved in [1] that the Jacobson radical of every finitely generated PI-ring is nilpotent. For every commutative semigroup ring RS, it was shown in [11] that if J(R) is nil then J(RS) is nil. This result was generalized to all semigroup algebras satisfying polynomial identities in [15] (see [16, Chapter 21]). Further, it was proved in [12] that, for every normal band B, if J(R) is nilpotent, then J(RB) is nilpotent. A similar result for special band-graded rings was established in [13, Section 6]. Analogous theorems concerning nilpotency and local nilpotency were proved in [2] for rings graded by finite and locally finite semigroups.

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13

Abdollahi, Alireza. "Certain locally nilpotent varieties of groups." Bulletin of the Australian Mathematical Society 67, no. 1 (February 2003): 115–19. http://dx.doi.org/10.1017/s0004972700033578.

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Let c ≥ 0, d ≥ 2 be integers and be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that is locally nilpotent? We prove that if c ≤ 2d + 2d−1 − 3 then the variety is locally nilpotent and we reduce the question of Gupta about the periodic groups in to the prime power exponent groups in this variety.

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14

Matveev, D. A. "Commuting homogeneous locally nilpotent derivations." Sbornik: Mathematics 210, no. 11 (November 2019): 1609–32. http://dx.doi.org/10.1070/sm9132.

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15

Wehrfritz, B. A. F. "Locally nilpotent skew linear groups." Proceedings of the Edinburgh Mathematical Society 29, no. 1 (February 1986): 101–13. http://dx.doi.org/10.1017/s0013091500017466.

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Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

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16

El Kahoui, M’hammed. "Subresultants and locally nilpotent derivations." Linear Algebra and its Applications 380 (March 2004): 253–61. http://dx.doi.org/10.1016/j.laa.2003.11.004.

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17

Finston, David R., and Sebastian Walcher. "Centralizers of locally nilpotent derivations." Journal of Pure and Applied Algebra 120, no. 1 (July 1997): 39–49. http://dx.doi.org/10.1016/s0022-4049(96)00064-3.

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18

Boudi, Nadia, and Martin Mathieu. "Locally quasi-nilpotent elementary operators." Operators and Matrices, no. 3 (2014): 785–98. http://dx.doi.org/10.7153/oam-08-44.

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19

Tanaka, Mikiya. "Locally nilpotent derivations on modules." Journal of Mathematics of Kyoto University 49, no. 1 (2009): 131–59. http://dx.doi.org/10.1215/kjm/1248983033.

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20

Fay, Temple H., and Gary L. Walls. "Categorically compact locally nilpotent groups." Communications in Algebra 18, no. 10 (January 1990): 3423–35. http://dx.doi.org/10.1080/00927879008824083.

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21

Robinson, Derek J. S. "Cohomology of locally nilpotent groups." Journal of Pure and Applied Algebra 48, no. 1-2 (September 1987): 281–300. http://dx.doi.org/10.1016/0022-4049(87)90116-2.

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22

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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23

Glöckner, Helge, and George A. Willis. "Locally pro-p contraction groups are nilpotent." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 781 (October 16, 2021): 85–103. http://dx.doi.org/10.1515/crelle-2021-0050.

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Abstract The authors have shown previously that every locally pro-p contraction group decomposes into the direct product of a p-adic analytic factor and a torsion factor. It has long been known that p-adic analytic contraction groups are nilpotent. We show here that the torsion factor is nilpotent too, and hence that every locally pro-p contraction group is nilpotent.

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24

ARIKAN, AHMET, and NADIR TRABELSI. "ON GROUPS WHOSE PROPER SUBGROUPS ARE CHERNIKOV-BY-BAER OR (PERIODIC DIVISIBLE ABELIAN)-BY-BAER." Journal of Algebra and Its Applications 12, no. 06 (May 9, 2013): 1350015. http://dx.doi.org/10.1142/s0219498813500151.

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If 𝔛 is a class of groups, then a group G is called a minimal non-𝔛-group if it is not an 𝔛-group but all of its proper subgroups belong to 𝔛. In this paper we prove that locally graded minimal non-(Chernikov-by-nilpotent)-groups are precisely minimal non-nilpotent-groups without maximal subgroups and that locally graded minimal non-(Chernikov-by-Baer)-groups are locally finite and coincide with the normal closure of an element. We also prove that an infinite locally graded minimal non-((periodic divisible abelian)-by-Baer)-group G is an imperfect locally nilpotent p-group, for some prime p, and there is an element a in G such that G = 〈a〉G.

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25

Santos Filho, G., L. Murakami, and I. Shestakov. "Locally finite coalgebras and the locally nilpotent radical I." Linear Algebra and its Applications 621 (July 2021): 235–53. http://dx.doi.org/10.1016/j.laa.2021.03.023.

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26

Höfling, Burkhard. "Subgroups of locally finite products of locally nilpotent groups." Glasgow Mathematical Journal 41, no. 3 (October 1999): 323–43. http://dx.doi.org/10.1017/s0017089599000294.

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27

Franciosi, Silvana, Francesco de Giovanni, and Yaroslav P. Sysak. "On locally finite groups factorized by locally nilpotent subgroups." Journal of Pure and Applied Algebra 106, no. 1 (January 1996): 45–56. http://dx.doi.org/10.1016/0022-4049(95)00004-6.

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28

Dixon, Martyn R., and Leonid A. Kurdachenko. "Locally nilpotent groups with the maximum condition on non-nilpotent subgroups." Glasgow Mathematical Journal 43, no. 1 (January 2001): 85–102. http://dx.doi.org/10.1017/s0017089501010072.

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29

Zhang, Zhirang, and Jiachao Li. "On Local Nilpotency of the Normal Subgroups of a Group." Algebra Colloquium 23, no. 03 (June 20, 2016): 531–40. http://dx.doi.org/10.1142/s1005386716000511.

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A group G is said to have property μ whenever N is a non-locally nilpotent normal subgroup of G, there is a finite non-nilpotent G-quotient of N. FC-groups and groups with property ν satisfy property μ, where a group G is said to have property ν if every non-nilpotent normal subgroup of G has a finite non-nilpotent G-quotient. HP(G) is the Hirsch-Plotkin radical of G, and Φf(G) is the intersection of all the maximal subgroups of finite index in G (here Φf(G)=G if no such maximal subgroups exist). It is shown that a group G has property μ if and only if HP(G/Φf(G))=HP(G)/Φf(G) and that the class of groups with property ν is a proper subclass of that of groups with property μ. Also, the structure of the normal subgroups of a group: nilpotency with the minimal condition, is investigated.

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30

Arenas, Manuel, and Alicia Labra. "Birrepresentations in a locally nilpotent variety." Proyecciones (Antofagasta) 33, no. 1 (March 2014): 123–32. http://dx.doi.org/10.4067/s0716-09172014000100009.

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31

Asar, A. O. "Barely Transitive Locally Nilpotent p -Groups." Journal of the London Mathematical Society 61, no. 1 (February 2000): 315–18. http://dx.doi.org/10.1112/s0024610799008121.

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32

Wehrfritz, B. A. F. "Locally Nilpotent Finitary Skew Linear Groups." Journal of the London Mathematical Society 50, no. 2 (October 1994): 323–40. http://dx.doi.org/10.1112/jlms/50.2.323.

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33

El Kahoui, M’hammed, and Mustapha Ouali. "Locally nilpotent derivations of factorial domains." Journal of Algebra and Its Applications 18, no. 12 (November 3, 2019): 1950222. http://dx.doi.org/10.1142/s0219498819502220.

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Let [Formula: see text] be factorial domains containing [Formula: see text]. In this paper, we give a criterion, in terms of locally nilpotent derivations, for [Formula: see text] to be [Formula: see text]-isomorphic to [Formula: see text], where [Formula: see text] is nonzero and [Formula: see text]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc. 145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent [Formula: see text]-derivations of the polynomial ring in two variables over [Formula: see text].

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34

Asar, A. O. "Barely Transitive Locally Nilpotent P-Groups." Journal of the London Mathematical Society 55, no. 2 (April 1997): 357–62. http://dx.doi.org/10.1112/s0024610797004845.

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35

Wehrfritz, B. A. F. "Locally nilpotent skew linear groups II." Proceedings of the Edinburgh Mathematical Society 30, no. 3 (October 1987): 423–26. http://dx.doi.org/10.1017/s001309150002681x.

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Our paper [6] studied in some depth certain locally nilpotent skew linear groups, but our conclusions there left some obvious gaps. By means of a trick, which now seems obvious, but then did not, we are able to tidy up the situation very satisfactorily. This present paper should be viewed as a follow up to [6]. In particular we do not repeat the motivation, basic definitions and references to related work given here.The following was conjectured in [6], where substantial steps were taken towards its solution.

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36

Grzeszczuk, Piotr. "Locally nilpotent skew extensions of rings." Journal of Pure and Applied Algebra 224, no. 9 (September 2020): 106360. http://dx.doi.org/10.1016/j.jpaa.2020.106360.

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37

Willis, G. "Totally disconnected, nilpotent, locally compact groups." Bulletin of the Australian Mathematical Society 55, no. 1 (February 1997): 143–46. http://dx.doi.org/10.1017/s0004972700030604.

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It is shown that, if G is a totally disconnected, compactly generated and nilpotent locally compact group, then it has a base of neighbourhoods of the identity consisting of compact, open, normal subgroups. An example is given showing that the hypothesis that G be compactly generated is necessary.

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38

Fay, Temple H., and Gary L. Walls. "Categorically compact locally nilpotent groups:a corrigendum." Communications in Algebra 20, no. 4 (January 1992): 1019–22. http://dx.doi.org/10.1080/00927879208824388.

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39

Hai, Bui Xuan, and Nguyen Van Thin. "On Locally Nilpotent Subgroups ofGL1(D)." Communications in Algebra 37, no. 2 (February 12, 2009): 712–18. http://dx.doi.org/10.1080/00927870802255287.

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40

Pakovich, F. B. "Locally nilpotent derivations of polynomial rings." Mathematical Notes 58, no. 2 (August 1995): 890–91. http://dx.doi.org/10.1007/bf02304113.

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41

Ferrero, Miguel, Yves Lequain, and Andrzej Nowicki. "A note on locally nilpotent derivations." Journal of Pure and Applied Algebra 79, no. 1 (May 1992): 45–50. http://dx.doi.org/10.1016/0022-4049(92)90125-y.

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42

Iván, L. "On not locally nilpotent Černikov groups." Periodica Mathematica Hungarica 16, no. 4 (December 1985): 245–49. http://dx.doi.org/10.1007/bf01848074.

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43

BEHN, ANTONIO, ALBERTO ELDUQUE, and ALICIA LABRA. "A CLASS OF LOCALLY NILPOTENT COMMUTATIVE ALGEBRAS." International Journal of Algebra and Computation 21, no. 05 (August 2011): 763–74. http://dx.doi.org/10.1142/s0218196711006455.

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This paper deals with the variety of commutative non associative algebras satisfying the identity [Formula: see text], γ ∈ K. In [3] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ ≠ -1, then any such algebra is locally nilpotent. Our results require characteristic ≠ 2, 3.

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44

El Kahoui, M’hammed, Najoua Essamaoui, and Mustapha Ouali. "A nilpotency criterion for derivations over reduced ℚ-algebras." International Journal of Algebra and Computation 31, no. 05 (May 18, 2021): 903–13. http://dx.doi.org/10.1142/s0218196721500429.

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Let [Formula: see text] be a reduced ring containing [Formula: see text] and let [Formula: see text] be commuting locally nilpotent derivations of [Formula: see text]. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form [Formula: see text], where [Formula: see text] are elements in [Formula: see text].

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45

BOVDI, V. "GROUP ALGEBRAS WHOSE UNIT GROUP IS LOCALLY NILPOTENT." Journal of the Australian Mathematical Society 109, no. 1 (May 7, 2020): 17–23. http://dx.doi.org/10.1017/s1446788719000557.

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We present a complete list of groups $G$ and fields $F$ for which: (i) the group of normalized units $V(FG)$ of the group algebra $FG$ is locally nilpotent; (ii) the set of nontrivial nilpotent elements of $FG$ is finite and nonempty, and $V(FG)$ is an Engel group.

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46

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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47

BERGEN, JEFFREY, and PIOTR GRZESZCZUK. "GK DIMENSION AND LOCALLY NILPOTENT SKEW DERIVATIONS." Glasgow Mathematical Journal 57, no. 3 (December 18, 2014): 555–67. http://dx.doi.org/10.1017/s0017089514000482.

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AbstractLet A be a domain over an algebraically closed field with Gelfand–Kirillov dimension in the interval [2,3). We prove that if A has two locally nilpotent skew derivations satisfying some natural conditions, then A must be one of five algebras. All five algebras are Noetherian, finitely generated, and have Gelfand–Kirillov dimension equal to 2. We also obtain some results comparing the Gelfand–Kirillov dimension of an algebra to its subring of invariants under a locally nilpotent skew derivation.

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48

DILMI, AMEL, and NADIR TRABELSI. "Groups whose proper subgroups are (locally finite)-by-(locally nilpotent)." Publicationes Mathematicae Debrecen 87, no. 1-2 (June 1, 2015): 209–19. http://dx.doi.org/10.5486/pmd.2015.7162.

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49

Ursul, M. I. "LOCALLY FINITE AND LOCALLY PROJECTIVELY NILPOTENT IDEALS OF TOPOLOGICAL RINGS." Mathematics of the USSR-Sbornik 53, no. 2 (February 28, 1986): 291–305. http://dx.doi.org/10.1070/sm1986v053n02abeh002922.

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50

Bruno, Brunella, and Susan E. Schuur. "On locally finite groups with a locally nilpotent maximal subgroup." Archiv der Mathematik 61, no. 3 (September 1993): 215–20. http://dx.doi.org/10.1007/bf01198716.

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