Some Triple Generations of the Lyons Sporadic Simple Group <i>Ly</i>

Malebogo Motalane; Zwelethemba Mpono

doi:doi:10.11648/j.acm.20231203.12

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Some Triple Generations of the Lyons Sporadic Simple Group Ly

Malebogo Motalane, Zwelethemba Mpono

Published in Applied and Computational Mathematics (Volume 12, Issue 3)

Received: 31 January 2023 Accepted: 16 May 2023 Published: 20 July 2023

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Abstract

According to the classification theorem, the Lyons group Ly is one of the 26 sporadic simple groups and has order 51765179004000000 = 2⁸•3⁷•5⁶•7•11•31•37•67. Since the completion of the classification of all finite simple groups, attention has now turned to other aspects e.g. generations of finite groups which entails determining elements which generate that finite group. As a finite nonabelian simple group, Ly can be generated by a minimum of two of its elements. We thus endeavour in the current study to determine some of the pairs of its elements of distinct prime orders from disctinct conjugacy classes with their product in another conjugacy class of elements of prime order which generate Ly and we call such generations triple generations. Triple generations of any finite group are used in the study of its symmetric genus, where the symmetric genus of a Hurwitz group G, of which Ly is known to be a Hurwitz group, is given by . If G is a finite group and lX, mY, nZ are conjugacy classes of elements of G, then G is said to be (l,m,n)-generated if with o(x) = l, o(y) = m and o(xy) = n. The number of distinct ordered pairs (x,y) satisfying such that xy = z, where is an arbitrary class representative, is denoted by ζ_G(lX,mY,nZ) and is known as the structure constant of the group algebra . The structure constants can be computed from the ordinary character table of G. We shall use the method of the structure constants to determine such generation and/or nongeneration. Thus the object in this paper is to study some of the triple generations of Ly which will thus pave the way towards the study of various combinations of three, four, five etc elements from distinct conjugacy classes which can generate Ly and lead to the ultimate determination of the maximum number of elements of Ly from distinct conjugacy classes of its elements which can generate Ly.

Published in	Applied and Computational Mathematics (Volume 12, Issue 3)
DOI	10.11648/j.acm.20231203.12
Page(s)	55-81
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

(p,q,r)-Generations, Maximal Subgroups, Primes, Structure Constants, Conjugacy Class Fusions

References

[1]	F. Ali, On the ranks of O’N and Ly, Discrete Appl. Math. 155, (2007), 394-399.
[2]	M. D. E. Conder, R. A. Wilson and A. J. Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (3), (1992), 653-663.
[3]	J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson Atlas of Finite Groups, Oxford University Press, New York, (1985).
[4]	S. Ganief and J. Moori, (2,3,t)-generations for the Janko group J₃, Comm. Algebra, 23(12), (1995), 4427-4437.
[5]	S. Ganief and J. Moori, (p,q,r)-generations for the smallest Conway group Co₃, J. Algebra, 188, (1997), 516-530.
[6]	M. S. Ganief, 2-Generations of the Sporadic Simple Groups, Ph.D Thesis, University on Natal, (1997).
[7]	The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.10; (2007). (http://www.gap- system.org)
[8]	J. F. Hurley and A. Rudvalis, Finite simple groups, Amer. Math. Monthly 84, (1977), 693-714.
[9]	C. Jansen, K. Lux, R. Parker and R. Wilson An Atlas of Brauer Characters, Oxford University Press Inc., New York, (1995).
[10]	J. Moori, (p,q,r)-generations for the Janko groups J₁ and J₂, Nova J. Algebra Geom. 2 (3), (1993), 277-285.
[11]	J. Moori, On the ranks of Janko groups J₁, J₂, J₃, article presented at the 41st Annual Congress of the South African Mathematical Society, Rand Afrikaans University, Johannesburg, (1998).
[12]	M. J. Motalane, Triple Generations of the Lyons Sporadic Simple Group, MSc Dissertation, University of South Africa, (2015).
[13]	Z Mpono, The Conjugacy Class Ranks of M₂₄, International J. Group Theory 6 (4), (2017), 53-58.
[14]	Z Mpono, Triple Generations and Connected Components of Brauer Graphs in M₂₄, Southeast Asian Bull. Math., 41 (2017), 65-89.
[15]	R. Lyons, Evidence for a new finite simple group, J. Algebra 20, (1972), 540-569.
[16]	T. T. Seretlo, Fischer Clifford Matrices and Character Tables of Certain Groups Associated with Simple Groups , HS and Ly, PhD Thesis, University of KwaZulu-Natal, (2011).
[17]	A. J. Woldar, Representing M₁₁, M₁₂, M₂₂ and M₂₃ on surfaces of least genus, Comm. Algebra 12 (1), 1990, 15-86.
[18]	A. J. Woldar, 3/2-generarion of the sporadic simple groups, Comm. Algebra, 22 (2), (1994), 675-685.

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APA Style

Malebogo Motalane, Zwelethemba Mpono. (2023). Some Triple Generations of the Lyons Sporadic Simple Group Ly. Applied and Computational Mathematics, 12(3), 55-81. https://doi.org/10.11648/j.acm.20231203.12

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Malebogo Motalane; Zwelethemba Mpono. Some Triple Generations of the Lyons Sporadic Simple Group Ly. Appl. Comput. Math. 2023, 12(3), 55-81. doi: 10.11648/j.acm.20231203.12

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AMA Style

Malebogo Motalane, Zwelethemba Mpono. Some Triple Generations of the Lyons Sporadic Simple Group Ly. Appl Comput Math. 2023;12(3):55-81. doi: 10.11648/j.acm.20231203.12

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@article{10.11648/j.acm.20231203.12,
  author = {Malebogo Motalane and Zwelethemba Mpono},
  title = {Some Triple Generations of the Lyons Sporadic Simple Group Ly},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {3},
  pages = {55-81},
  doi = {10.11648/j.acm.20231203.12},
  url = {https://doi.org/10.11648/j.acm.20231203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231203.12},
  abstract = {According to the classification theorem, the Lyons group Ly is one of the 26 sporadic simple groups and has order 51765179004000000 = 28•37•56•7•11•31•37•67. Since the completion of the classification of all finite simple groups, attention has now turned to other aspects e.g. generations of finite groups which entails determining elements which generate that finite group. As a finite nonabelian simple group, Ly can be generated by a minimum of two of its elements. We thus endeavour in the current study to determine some of the pairs of its elements of distinct prime orders from disctinct conjugacy classes with their product in another conjugacy class of elements of prime order which generate Ly and we call such generations triple generations. Triple generations of any finite group are used in the study of its symmetric genus, where the symmetric genus of a Hurwitz group G, of which Ly is known to be a Hurwitz group, is given by . If G is a finite group and lX, mY, nZ are conjugacy classes of elements of G, then G is said to be (l,m,n)-generated if  with o(x) = l, o(y) = m and o(xy) = n. The number of distinct ordered pairs (x,y) satisfying  such that xy = z, where  is an arbitrary class representative, is denoted by ζG(lX,mY,nZ) and is known as the structure constant of the group algebra . The structure constants can be computed from the ordinary character table of G. We shall use the method of the structure constants to determine such generation and/or nongeneration. Thus the object in this paper is to study some of the triple generations of Ly which will thus pave the way towards the study of various combinations of three, four, five etc elements from distinct conjugacy classes which can generate Ly and lead to the ultimate determination of the maximum number of elements of Ly from distinct conjugacy classes of its elements which can generate Ly.},
 year = {2023}
}

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TY  - JOUR
T1  - Some Triple Generations of the Lyons Sporadic Simple Group Ly
AU  - Malebogo Motalane
AU  - Zwelethemba Mpono
Y1  - 2023/07/20
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231203.12
DO  - 10.11648/j.acm.20231203.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 55
EP  - 81
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231203.12
AB  - According to the classification theorem, the Lyons group Ly is one of the 26 sporadic simple groups and has order 51765179004000000 = 28•37•56•7•11•31•37•67. Since the completion of the classification of all finite simple groups, attention has now turned to other aspects e.g. generations of finite groups which entails determining elements which generate that finite group. As a finite nonabelian simple group, Ly can be generated by a minimum of two of its elements. We thus endeavour in the current study to determine some of the pairs of its elements of distinct prime orders from disctinct conjugacy classes with their product in another conjugacy class of elements of prime order which generate Ly and we call such generations triple generations. Triple generations of any finite group are used in the study of its symmetric genus, where the symmetric genus of a Hurwitz group G, of which Ly is known to be a Hurwitz group, is given by . If G is a finite group and lX, mY, nZ are conjugacy classes of elements of G, then G is said to be (l,m,n)-generated if  with o(x) = l, o(y) = m and o(xy) = n. The number of distinct ordered pairs (x,y) satisfying  such that xy = z, where  is an arbitrary class representative, is denoted by ζG(lX,mY,nZ) and is known as the structure constant of the group algebra . The structure constants can be computed from the ordinary character table of G. We shall use the method of the structure constants to determine such generation and/or nongeneration. Thus the object in this paper is to study some of the triple generations of Ly which will thus pave the way towards the study of various combinations of three, four, five etc elements from distinct conjugacy classes which can generate Ly and lead to the ultimate determination of the maximum number of elements of Ly from distinct conjugacy classes of its elements which can generate Ly.
VL  - 12
IS  - 3
ER  -

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Author Information

Malebogo Motalane

Department of Mathematics and Applied Mathematics, Faculty of Science, University of Limpopo, Sovenga, South Africa
Zwelethemba Mpono

Department of Mathematical Sciences, College of Science, Engineering and Technology, University of South Africa, Pretoria, South Africa

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Malebogo Motalane, Zwelethemba Mpono. (2023). Some Triple Generations of the Lyons Sporadic Simple Group Ly. Applied and Computational Mathematics, 12(3), 55-81. https://doi.org/10.11648/j.acm.20231203.12

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ACS Style

Malebogo Motalane; Zwelethemba Mpono. Some Triple Generations of the Lyons Sporadic Simple Group Ly. Appl. Comput. Math. 2023, 12(3), 55-81. doi: 10.11648/j.acm.20231203.12

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AMA Style

Malebogo Motalane, Zwelethemba Mpono. Some Triple Generations of the Lyons Sporadic Simple Group Ly. Appl Comput Math. 2023;12(3):55-81. doi: 10.11648/j.acm.20231203.12

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@article{10.11648/j.acm.20231203.12,
  author = {Malebogo Motalane and Zwelethemba Mpono},
  title = {Some Triple Generations of the Lyons Sporadic Simple Group Ly},
  journal = {Applied and Computational Mathematics},
  volume = {12},
  number = {3},
  pages = {55-81},
  doi = {10.11648/j.acm.20231203.12},
  url = {https://doi.org/10.11648/j.acm.20231203.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231203.12},
  abstract = {According to the classification theorem, the Lyons group Ly is one of the 26 sporadic simple groups and has order 51765179004000000 = 28•37•56•7•11•31•37•67. Since the completion of the classification of all finite simple groups, attention has now turned to other aspects e.g. generations of finite groups which entails determining elements which generate that finite group. As a finite nonabelian simple group, Ly can be generated by a minimum of two of its elements. We thus endeavour in the current study to determine some of the pairs of its elements of distinct prime orders from disctinct conjugacy classes with their product in another conjugacy class of elements of prime order which generate Ly and we call such generations triple generations. Triple generations of any finite group are used in the study of its symmetric genus, where the symmetric genus of a Hurwitz group G, of which Ly is known to be a Hurwitz group, is given by . If G is a finite group and lX, mY, nZ are conjugacy classes of elements of G, then G is said to be (l,m,n)-generated if  with o(x) = l, o(y) = m and o(xy) = n. The number of distinct ordered pairs (x,y) satisfying  such that xy = z, where  is an arbitrary class representative, is denoted by ζG(lX,mY,nZ) and is known as the structure constant of the group algebra . The structure constants can be computed from the ordinary character table of G. We shall use the method of the structure constants to determine such generation and/or nongeneration. Thus the object in this paper is to study some of the triple generations of Ly which will thus pave the way towards the study of various combinations of three, four, five etc elements from distinct conjugacy classes which can generate Ly and lead to the ultimate determination of the maximum number of elements of Ly from distinct conjugacy classes of its elements which can generate Ly.},
 year = {2023}
}

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TY  - JOUR
T1  - Some Triple Generations of the Lyons Sporadic Simple Group Ly
AU  - Malebogo Motalane
AU  - Zwelethemba Mpono
Y1  - 2023/07/20
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231203.12
DO  - 10.11648/j.acm.20231203.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 55
EP  - 81
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231203.12
AB  - According to the classification theorem, the Lyons group Ly is one of the 26 sporadic simple groups and has order 51765179004000000 = 28•37•56•7•11•31•37•67. Since the completion of the classification of all finite simple groups, attention has now turned to other aspects e.g. generations of finite groups which entails determining elements which generate that finite group. As a finite nonabelian simple group, Ly can be generated by a minimum of two of its elements. We thus endeavour in the current study to determine some of the pairs of its elements of distinct prime orders from disctinct conjugacy classes with their product in another conjugacy class of elements of prime order which generate Ly and we call such generations triple generations. Triple generations of any finite group are used in the study of its symmetric genus, where the symmetric genus of a Hurwitz group G, of which Ly is known to be a Hurwitz group, is given by . If G is a finite group and lX, mY, nZ are conjugacy classes of elements of G, then G is said to be (l,m,n)-generated if  with o(x) = l, o(y) = m and o(xy) = n. The number of distinct ordered pairs (x,y) satisfying  such that xy = z, where  is an arbitrary class representative, is denoted by ζG(lX,mY,nZ) and is known as the structure constant of the group algebra . The structure constants can be computed from the ordinary character table of G. We shall use the method of the structure constants to determine such generation and/or nongeneration. Thus the object in this paper is to study some of the triple generations of Ly which will thus pave the way towards the study of various combinations of three, four, five etc elements from distinct conjugacy classes which can generate Ly and lead to the ultimate determination of the maximum number of elements of Ly from distinct conjugacy classes of its elements which can generate Ly.
VL  - 12
IS  - 3
ER  -

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