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Journal articles on the topic 'Soluble groups][Automorphisms'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 February 2022

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1

Flavell, Paul. "Automorphisms of soluble groups." Proceedings of the London Mathematical Society 112, no. 4 (April 2016): 623–50. http://dx.doi.org/10.1112/plms/pdw005.

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2

Gromadzki, Grzegorz. "On Soluble Groups of Automorphism of Riemann Surfaces." Canadian Mathematical Bulletin 34, no. 1 (March 1, 1991): 67–73. http://dx.doi.org/10.4153/cmb-1991-011-x.

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AbstractLet G be a soluble group of derived length 3. We show in this paper that if G acts as an automorphism group on a compact Riemann surface of genus g ≠ 3,5,6,10 then it has at most 24(g — 1) elements. Moreover, given a positive integer n we show the existence of a Riemann surface of genus g = n4 + 1 that admits such a group of automorphisms of order 24(g — 1), whilst a surface of specified genus can admit such a group of automorphisms of order 48(g — 1), 40(g — 1), 30(g — 1) and 36(g — 1) respectively.

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3

Endimioni, Gérard. "Polynomial Automorphisms of Soluble Groups." Communications in Algebra 37, no. 10 (October 9, 2009): 3388–400. http://dx.doi.org/10.1080/00927870802502837.

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4

Smith, Michael J. "Computing automorphisms of finite soluble groups." Bulletin of the Australian Mathematical Society 53, no. 1 (February 1996): 169–71. http://dx.doi.org/10.1017/s0004972700016841.

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5

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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6

Shumyatsky, P. "Involutory Automorphisms of Locally Soluble Periodic Groups." Journal of Algebra 155, no. 1 (February 1993): 36–43. http://dx.doi.org/10.1006/jabr.1993.1030.

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7

Bastos, Raimundo, Alex C. Dantas, and Emerson de Melo. "Soluble groups with few orbits under automorphisms." Geometriae Dedicata 209, no. 1 (March 17, 2020): 119–23. http://dx.doi.org/10.1007/s10711-020-00525-7.

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8

Wehrfritz, B. A. F. "Almost fixed-point-free automorphisms of soluble groups." Journal of Pure and Applied Algebra 215, no. 5 (May 2011): 1112–15. http://dx.doi.org/10.1016/j.jpaa.2010.07.017.

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9

Gromadzki, G. "On soluble groups of automorphisms of nonorientable Klein surfaces." Fundamenta Mathematicae 141, no. 3 (1992): 215–27. http://dx.doi.org/10.4064/fm-141-3-215-227.

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10

Wehrfritz, Bertram A. F. "On soluble groups of module automorphisms of finite rank." Czechoslovak Mathematical Journal 67, no. 3 (August 9, 2017): 809–18. http://dx.doi.org/10.21136/cmj.2017.0193-16.

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11

Timoshenko, E. I. "Normal automorphisms of a soluble product of abelian groups." Siberian Mathematical Journal 56, no. 1 (January 2015): 191–98. http://dx.doi.org/10.1134/s0037446615010188.

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12

LIU, HeGuo, and Tao XU. "On almost regular automorphisms of residually finite minimax soluble groups." SCIENTIA SINICA Mathematica 42, no. 12 (December 1, 2012): 1237–50. http://dx.doi.org/10.1360/012012-325.

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13

Wehrfritz, Bertram A. F. "Automorphisms of Finite Order of Soluble Groups of Finite Rank." Studia Scientiarum Mathematicarum Hungarica 58, no. 1 (April 14, 2021): 19–31. http://dx.doi.org/10.1556/012.2021.58.1.1486.

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We study the effect on sections of a soluble-by-finite group G of finite rank of an almost fixed-point-free automorphism φ of G of finite order. We also elucidate the structure of G if φ has order 4 and if G is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

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14

Gromadzki, Grzegorz, and Colin MacLachlan. "Supersoluble groups of automorphisms of compact Riemann surfaces." Glasgow Mathematical Journal 31, no. 3 (September 1989): 321–27. http://dx.doi.org/10.1017/s0017089500007886.

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Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

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15

Kai-Nah, Cheng. "Finite groups admitting fixed point free automorphisms of order pq." Proceedings of the Edinburgh Mathematical Society 30, no. 1 (February 1987): 51–56. http://dx.doi.org/10.1017/s0013091500017958.

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By the results of Rickman [7] and Ralston [6], a finite group G admitting a fixed point free automorphism α of order pq, where p and q are primes, is soluble. If p = q, then |G| is necessarily coprime to |α|, and it follows from Berger [1] that G has Fitting height at most 2, the composition length of <α>. The purpose of this paper is to prove a corresponding result in the case when p≠q.

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16

Khukhro, E. I. "On finite soluble groups with almost fixed-point-free automorphisms of noncoprime order." Siberian Mathematical Journal 56, no. 3 (May 2015): 541–48. http://dx.doi.org/10.1134/s0037446615030179.

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17

ERSOY, KIVANÇ. "CENTRALIZERS OF p-SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS." Glasgow Mathematical Journal 62, no. 1 (February 21, 2019): 183–86. http://dx.doi.org/10.1017/s001708951900003x.

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AbstractIn Ersoy et al. [J. Algebra481 (2017), 1–11], we have proved that if G is a locally finite group with an elementary abelian p-subgroup A of order strictly greater than p2 such that CG(A) is Chernikov and for every non-identity α ∈ A the centralizer CG(α) does not involve an infinite simple group, then G is almost locally soluble. This result is a consequence of another result proved in Ersoy et al. [J. Algebra481 (2017), 1–11], namely: if G is a simple locally finite group with an elementary abelian group A of automorphisms acting on it such that the order of A is greater than p2, the centralizer CG(A) is Chernikov and for every non-identity α ∈ A the set of fixed points CG(α) does not involve an infinite simple groups then G is finite. In this paper, we improve this result about simple locally finite groups: Indeed, suppose that G is a simple locally finite group, consider a finite non-abelian subgroup P of automorphisms of exponent p such that the centralizer CG(P) is Chernikov and for every non-identity α ∈ P the set of fixed points CG(α) does not involve an infinite simple group. We prove that if Aut(G) has such a subgroup, then G ≅PSLp(k) where char k ≠ p and P has a subgroup Q of order p2 such that CG(P) = Q.

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18

Lichtman, A. I. "Automorphism Groups of Free Soluble Groups." Journal of Algebra 174, no. 1 (May 1995): 132–49. http://dx.doi.org/10.1006/jabr.1995.1120.

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19

Laffey, Thomas J., and Desmond MacHale. "Automorphism orbits of finite groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40, no. 2 (April 1986): 253–60. http://dx.doi.org/10.1017/s1446788700027221.

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AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

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20

Hegarty, P. V. "Soluble Groups with an Automorphism Inverting Many Elements." Mathematical Proceedings of the Royal Irish Academy 105, no. 1 (January 1, 2005): 59–73. http://dx.doi.org/10.3318/pria.2005.105.1.59.

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21

HUANG, ZHAOHONG, JIANGMIN PAN, SUYUN DING, and ZHE LIU. "AUTOMORPHISM GROUPS OF SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS." Bulletin of the Australian Mathematical Society 93, no. 2 (November 11, 2015): 238–47. http://dx.doi.org/10.1017/s0004972715001197.

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Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$-power-free order.

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22

Derrick, A. J. "Groups with no nontrivial linear representations." Bulletin of the Australian Mathematical Society 50, no. 1 (August 1994): 1–11. http://dx.doi.org/10.1017/s0004972700009503.

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We study the class of groups having no nontrivial linear representations over certain fields. After showing the class to be closed under perfect extensions with locally soluble kernel, we expand considerably the number of acyclic groups known to be in the class, by application to both binate groups and the acyclic automorphism groups of de la Harpe and McDuff.

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23

ERSOY, KIVANÇ, ANTONIO TORTORA, and MARIA TOTA. "ON GROUPS WITH ALL SUBGROUPS SUBNORMAL OR SOLUBLE OF BOUNDED DERIVED LENGTH." Glasgow Mathematical Journal 56, no. 1 (August 13, 2013): 221–27. http://dx.doi.org/10.1017/s0017089513000190.

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AbstractIn this paper we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.

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24

Brookes, C. J. B., and J. R. J. Groves. "Some infinite soluble groups, their modules, and the arithmeticity of associated automorphism groups." Journal of Algebra 283, no. 2 (January 2005): 485–504. http://dx.doi.org/10.1016/j.jalgebra.2004.01.018.

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25

Parrott, David. "Finite groups which admit a fixed-point-free automorphism group isomorphic to S3." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 48, no. 3 (June 1990): 384–96. http://dx.doi.org/10.1017/s1446788700029931.

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26

Liu, Hailin. "Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer." Open Mathematics 17, no. 1 (May 30, 2019): 513–18. http://dx.doi.org/10.1515/math-2019-0041.

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Abstract A Cayley graph Γ is said to be arc-transitive if its full automorphism group AutΓ is transitive on the arc set of Γ. In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of Frobenius groups with soluble vertex stabilizer.

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27

Khukhro, E. I., N. Yu Makarenko, and P. Shumyatsky. "Finite Groups and Lie Rings with an Automorphism of Order 2n." Proceedings of the Edinburgh Mathematical Society 60, no. 2 (June 15, 2016): 391–412. http://dx.doi.org/10.1017/s0013091516000225.

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AbstractSuppose that a finite groupGadmits an automorphismof order 2nsuch that the fixed-point subgroupof the involutionis nilpotent of classc. Letm=) be the number of fixed points of. It is proved thatGhas a characteristic soluble subgroup of derived length bounded in terms ofn,cwhose index is bounded in terms ofm,n,c. A similar result is also proved for Lie rings.

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28

Hartley, Brian, and Volker Turau. "Finite soluble groups admitting an automorphism of prime power order with few fixed points." Mathematical Proceedings of the Cambridge Philosophical Society 102, no. 3 (November 1987): 431–41. http://dx.doi.org/10.1017/s0305004100067487.

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Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of theCONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.

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29

Khukhro, E. I. "Finite soluble and niplotent groups with a restriction on the rank of the centralizer of an automorphism of prime order." Siberian Mathematical Journal 41, no. 2 (April 2000): 373–88. http://dx.doi.org/10.1007/bf02674608.

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30

Frati, Marco. "Unipotent automorphisms of soluble groups with finite Prüfer rank." Journal of Group Theory 17, no. 3 (May 1, 2014). http://dx.doi.org/10.1515/jgt-2013-0041.

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