ANALYSING THE COMMUTING GRAPHS FOR ELEMENTS OF ORDER 3 IN MATHIEU GROUPS
DOI:
https://doi.org/10.61841/0wejp560Keywords:
Sporadic groups, commuting graph, diameter, cliquesAbstract
Assume that G is a finite group and X is a subset of G. The commuting graph is denoted by С(G,X) which has a set of vertices X with two distinct vertices x, y Î X being connected together on the condition of xy = yx. In this paper, computational approaches applied to investigate the structure of commuting graphs Ϲ(G,X) when G is one of the Mathieu groups along with X a G-conjugacy class for elements of order 3. We will pay particular attention to analyze the discs structure and determinate the diameters, girths and the clique number for these graphs.
Downloads
References
1. R.Brauer and K. A .Fowler, “On Groups of Even Order”,Ann. of Math., Vol. 62, no. 3.565-583, 1955.
2. S.Rapinchuk, Y.Segev and M.Seitz, “Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable”, J. Amer. Math. Soc., 15(4) 929-978, 2002.
3. C.Bates, D.Bundy, S.Hart, and P.Rowley,”Commuting involution graphs for sporadic simple groups”, J. Algebra,316(2): 849-868, 2007.
4. C.Bates, D.Bundy, S.Hart, and P.Rowley,”Commuting involution graphs for finite Coxeter groups”, J. Group Theory,6(4):461-476,2003.
5. C.Bates, D.Bundy, S.Hart, and P.Rowley,”Commuting involution graphs for symmetric groups”, J. Algebra, 266(1) .133-153, 2003.
6. C.Bates, D.Bundy, S.Hart, and P.Rowley,”Commuting involution graphs in special linear groups”, Comm. Algebra, 32(11)4179-4196,2004.
7. A.Nawawi, and P.Rowley, “On commuting graphs for elements of order 3 in symmetric groups”, PhD thesis,University of Manchester.2012.
8. . J. Brinkman, W. Bosma, J. Cannon and C. Playoust,”The Magma algebra system. I.The user language”,
J. Symbolic Comput. 24(3-4) 235-265, 1997.
9. The GAP Group.,”GAP Groups, Algorithms, and Programming”,. Version 4.4.12, http://www.gap-system.org, 2008.
10. J. Tripp, I. Suleiman, S. Rogers R. Parker, S. Norton, S. Nickerson, S. Linton,
11. J. Bray, A. Wilson and P. Walsh, “A world wide web atlas of group representations”.,
http://brauer.maths.qmul.ac.uk/Atlas/v3/.
12. H. Conway, R. T. Curtis, S. P .Norton and R. A. Parker,” ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups”, Oxford. Clarendon press.1985.
13. .I.M. Isaacs,”Character theory of finite Simple group. New York. Academic Press”, 1976.
14. D.Gorenstein,“Finite Groups. 2nd ed, Chelsea, Chelsea Publishing Company”,1980.
15. C. Cedillo, R. MacKinney-Romero, M.A. Pizaa, I.A. Robles and R. Villarroel-Flores,”Yet Another Graph System,YAGS”, Version 0.0.5. http://xamanek.izt.uam.mx/yags/. 2019.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
You are free to:
- Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
- Adapt — remix, transform, and build upon the material for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
Under the following terms:
- Attribution — You must give appropriate credit , provide a link to the license, and indicate if changes were made . You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
Notices:
You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation .
No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material.
