Relevant bibliographies by topics / Wielandt subgroup

Academic literature on the topic 'Wielandt subgroup'

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Published: 4 June 2021
Last updated: 28 January 2023

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Journal articles on the topic "Wielandt subgroup"

1

Cossey, John. "The Wielandt subgroup of a polycyclic group." Glasgow Mathematical Journal 33, no. 2 (May 1991): 231–34. http://dx.doi.org/10.1017/s0017089500008260.

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The purpose of this paper is to establish some basic properties of the Wielandt subgroup of a polycyclic group. The Wielandt subgroup of a group G is defined to be the intersection of the normalisers of all the subnormal subgroups of G and is denoted by ω(G). In 1958 Wielandt [9] showed that any minimal normal subgroup with the minimum condition on subnormal subgroups is contained in the Wielandt subgroup: it follows that the Wielandt subgroup of an artinian group is nontrivial. In contrast, the Wielandt subgroup of a polycyclic group can be trivial; an easy example is given by the infinite dihedral group. We will show that the Wielandt subgroup of a polycyclic group is close to being central.
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Ormerod, Elizabeth A. "Groups of Wielandt length two." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 2 (September 1991): 229–44. http://dx.doi.org/10.1017/s0305004100070304.

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The Wielandt subgroup ω(G) of a group G is the subgroup of elements that normalize every subnormal subgroup of G. This subgroup, now named for Wielandt, was introduced by him in 1958 [15]. For a finite non-trivial group the Wielandt subgroup is always a non-trivial, characteristic subgroup. Thus it is possible to define the ascending Wielandt series for a finite group G which terminates at the group. Write ω0(G) = 1, and for i ≥ 1, ωi(G)/ωi–1(G) = ω(G/ωi–1(G)). The smallest n such that ωn(G) = G is called the Wielandt length of G, and the class of groups of Wielandt length at most n is denoted by . From the definition it follows that is closed under homomorphic images and taking normal subgroups. Nilpotent groups in are also closed under taking subgroups.
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Bryce, R. A., John Cossey, and E. A. Ormerod. "A note on p-groups with power automorphisms." Glasgow Mathematical Journal 34, no. 3 (September 1992): 327–32. http://dx.doi.org/10.1017/s0017089500008892.

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Let G be a group. The norm, or Kern of G is the subgroup of elements of G which normalize every subgroup of the group. This idea was introduced in 1935 by Baer [1, 2], who delineated the basic properties of the norm. A related concept is the subgroup introduced by Wielandt [10] in 1958, and now named for him. The Wielandt subgroup of a group G is the subgroup of elements normalizing every subnormal subgroup of G. In the case of finite nilpotent groups these two concepts coincide, of course, since all subgroups of a finite nilpotent group are subnormal. Of late the Wielandt subgroup has been widely studied, and the name tends to be the more used, even in the finite nilpotent context when, perhaps, norm would be more natural. We denote the Wielandt subgroup of a group G by ω(G). The Wielandt series of subgroups ω1(G) is defined by: ω1(G) = ω(G) and for i ≥ 1, ωi+1(G)/ ω(G) = ω(G/ωi, (G)). The subgroups of the upper central series we denote by ζi(G).
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Ormerod, Elizabeth A. "The Wielandt subgroup of metacyclic p-groups." Bulletin of the Australian Mathematical Society 42, no. 3 (December 1990): 499–510. http://dx.doi.org/10.1017/s0004972700028665.

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The Wielandt subgroup is the intersection of the normalisers of all the subnormal subgroups of a group. For a finite group it is a non-trivial characteristic subgroup, and this makes it possible to define an ascending normal series terminating at the group. This series is called the Wielandt series and its length determines the Wielandt length of the group. In this paper the Wielandt subgroup of a metacyclic p–group is identified, and using this information it is shown that if a metacyclic p–group has Wielandt length n, its nilpotency class is n or n + 1.
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Ali, A. "On the Wielandt length of a finite supersoluble group." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 6 (December 2000): 1217–26. http://dx.doi.org/10.1017/s0308210500000640.

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We develop a technique for calculating the Wielandt subgroup of a semidirect product of two finite groups of coprime order. We apply this technique to calculate the Wielandt length of a supersoluble group in terms of the Wielandt lengths of its Sylow subgroups (for small Wielandt lengths) and in terms of the nilpotency classes of its Sylow subgroups.
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Brandl, Rolf, Silvana Franciosi, and Francesco de Giovanni. "On the Wielandt subgroup of infinite soluble groups." Glasgow Mathematical Journal 32, no. 2 (May 1990): 121–25. http://dx.doi.org/10.1017/s0017089500009149.

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The Wielandt subgroup w(G) of a group G is defined to be the intersection of the normalizers of all the subnormal subgroups of G. If G is a group satisfying the minimal condition on subnormal subgroups then Wielandt [10] showed that w(G) contains every minimal normal subgroup of G, and so contains the socle of G, and, later, Robinson [6] and Roseblade [9] proved that w(G) has finite index in G.
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Casolo, Carlo. "Soluble Groups with Finite Wielandt length." Glasgow Mathematical Journal 31, no. 3 (September 1989): 329–34. http://dx.doi.org/10.1017/s0017089500007898.

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The Wielandt subgroup ω(G) of a group G is defined to be the intersection of all normalizers of subnormal subgroups of G; the terms of the Wielandt series of G are defined, inductively, by putting ω0(G) = 1 and (ωn+1(G)/ωn(G) = ω(G/ωn(G)). If, for some integer n, ωn(G) = G, then G is said to have finite Wielandt length; the Wielandt length of G being the minimal n such that ωn(G) = G.
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Ormerod, Elizabeth A. "Some p-groups of Weilandt length three." Bulletin of the Australian Mathematical Society 58, no. 1 (August 1998): 121–36. http://dx.doi.org/10.1017/s0004972700032056.

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Finite p-groups of Wielandt length 1 are groups in which every subgroup is normal and are Dedekind groups. When the prime is odd therefore a finite p-group of Wielandt length 1 is Abelian. For an odd prime, a finite p-group of Wielandt length 2 has nilpotency class at most 3 and for such a goup to have class 3 there must be a 2-generator subgroup of this class. In this paper it is shown that for any prime p > 3 a finite p-group of Wielandt length 3 has nilpotency class at most 4, and for such a group to have class 4 there must be a 2-generator subgroup with this class. Two families of p-groups of Wielandt length 3 are described. One is a family of 3-generator groups with the property that each group modulo its Wielandt subgroup has class 2, the other is a family of 2-generator groups with the property that each group modulo its Wielandt subgroup has class 3.
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9

Beidleman, James C., Martyn R. Dixon, and Derek J. S. Robinson. "The Generalized Wielandt Subgroup of a Group." Canadian Journal of Mathematics 47, no. 2 (April 1, 1995): 246–61. http://dx.doi.org/10.4153/cjm-1995-012-7.

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AbstractThe intersection IW(G) of the normalizers of the infinite subnormal subgroups of a group G is a characteristic subgroup containing the Wielandt subgroup W(G) which we call the generalized Wielandt subgroup. In this paper we show that if G is infinite, then the structure of IW(G)/ W(G) is quite restricted, being controlled by a certain characteristic subgroup S(G). If S(G) is finite, then so is IW(G)/ W(G), whereas if S(G) is an infinite Prüfer-by-finite group, then IW(G)/W(G) is metabelian. In all other cases, IW(G) = W(G).
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Wetherell, C. J. T. "The wielandt series of metabelian groups." Bulletin of the Australian Mathematical Society 67, no. 2 (April 2003): 267–76. http://dx.doi.org/10.1017/s0004972700033736.

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The Wielandt subgroup of a group is the intersection of the normalisers of its subnormal subgroups. It is non-trivial in any finite group and thus gives rise to a series whose length provides a measure of the complexity of the group's subnormal structure. In this paper results of Ormerod concerning the interplay between the Wielandt series and upper central series of metabelian p-groups, p odd, are extended to the class of all odd order metabelian groups. These extensions are formulated in terms of a natural generalization of the upper central series which arises from Casolo's strong Wielandt subgroup, the intersection of the centralisers of a group's nilpotent subnormal sections.
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Dissertations / Theses on the topic "Wielandt subgroup"

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Wetherell, Chris, and chrisw@wintermute anu edu au. "Subnormal Structure of Finite Soluble Groups." The Australian National University. Faculty of Science, 2001. http://thesis.anu.edu.au./public/adt-ANU20020607.121248.

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The Wielandt subgroup, the intersection of normalizers of subnormal subgroups, is non-trivial in any finite group and thus gives rise to a series whose length is a measure of the complexity of a group's subnormal structure. Another measure, akin to the nilpotency class of nilpotent groups, arises from the strong Wielandt subgroup, the intersection of centralizers of nilpotent subnormal sections. This thesis begins an investigation into how these two invariants relate in finite soluble groups. ¶ Complete results are obtained for metabelian groups of odd order: the strong Wielandt length of such a group is at most one more than its Wielandt length, and this bound is best possible. Some progress is made in the wider class of groups with p-length 1 for all primes p. A conjecture for all finite soluble groups, which may be regarded as a subnormal analogue of the embedding of the Kern, is also considered.
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Amin, Fazli [Verfasser], and Bettina [Akademischer Betreuer] Eick. "Generalisations of the Wielandt subgroup / Fazli Amin ; Betreuer: Bettina Eick." Braunschweig : Technische Universität Braunschweig, 2016. http://d-nb.info/117581833X/34.

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Amin, Fazli Verfasser], and Bettina [Akademischer Betreuer] [Eick. "Generalisations of the Wielandt subgroup / Fazli Amin ; Betreuer: Bettina Eick." Braunschweig : Technische Universität Braunschweig, 2016. http://nbn-resolving.de/urn:nbn:de:gbv:084-16090114070.

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4

Wetherell, Chris. "Subnormal Structure of Finite Soluble Groups." Phd thesis, 2001. http://hdl.handle.net/1885/48016.

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The Wielandt subgroup, the intersection of normalizers of subnormal subgroups, is non-trivial in any finite group and thus gives rise to a series whose length is a measure of the complexity of a group's subnormal structure. Another measure, akin to the nilpotency class of nilpotent groups, arises from the strong Wielandt subgroup, the intersection of centralizers of nilpotent subnormal sections. This thesis begins an investigation into how these two invariants relate in finite soluble groups. ¶ Complete results are obtained for metabelian groups of odd order: the strong Wielandt length of such a group is at most one more than its Wielandt length, and this bound is best possible. Some progress is made in the wider class of groups with p-length 1 for all primes p. A conjecture for all finite soluble groups, which may be regarded as a subnormal analogue of the embedding of the Kern, is also considered.
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