The ranks of the classes of $A_{10}$

Document Type : Research Paper


School of Mathematical and Computer Sciences‎, ‎University of Limpopo (Turfloop)‎, ‎P‎. ‎Bag X1106‎, ‎Sovenga 0727‎, ‎South Africa.


‎Let $G $ be a finite group and $X$ be a conjugacy class of $G.$ The‎ ‎rank of $X$ in $G,$ denoted by $rank(G{:}X),$ is defined to‎ ‎be the minimal number of elements of $X$ generating $G.$ In this‎ ‎paper we establish the ranks of all the conjugacy classes of‎ ‎elements for simple alternating group $A_{10}$ using the structure‎ ‎constants method and other results established in‎ ‎[A.B.M‎. ‎Basheer and J‎. ‎Moori‎, ‎On the ranks of the alternating group $A_{n}$‎, ‎Bull‎. ‎Malays‎. ‎Math‎. ‎Sci‎. ‎Soc..


Main Subjects

F. Ali, On the ranks of ON and Ly, Discrete Appl. Math. 155 (2007) 394--399.
F. Ali, On the ranks of Fi22, Quaest. Math. 37 (2014) 591--600.
F. Ali and M.A.F. Ibrahim, On the ranks of Conway groups Co2 and Co3, J. Algebra Appl. 45 (2005) 557--565.
F. Ali and M.A.F. Ibrahim, On the ranks of HS and McL, Util. Math. 70 (2006) 187--195.
F. Ali and J. Moori, On the ranks of Janko groups J1; J2; J3 and J4, Quaest. Math. 31 (2008) 37--44.
A.B.M. Basheer and J. Moori, On the ranks of finite simple groups, Khayyam J. Math. 31 (2016) 18--24.
A.B.M. Basheer and J. Moori, On the ranks of the alternating group An, Bull. Malays. Math. Sci. Soc.
W. Bosma and J.J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of Sydeny, November 1994.
M.D.E. Conder, R.A. Wilson and A. J.Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992) 653--663.
J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
S. Ganief, 2-Generations of the Sporadic Simple Groups, PhD Thesis, University of Natal, South Africa, 1997.
S. Ganief and J. Moori, (p; q; r)-generations of the smallest Conway group Co3, J. Algebra 188 (1997) 516--530.
S. Ganief and J. Moori, 2-generations of the smallest Fischer group F22, Nova J. Math. Game Theory Algebra 6 (1997) 127--145.
The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.4.10; 2007,
J.I. Hall and L.H. Soicher, Presentations of some 3-transposition groups, Comm. Algebra 23 (1995) 2517--2559.
L. Di Martino, M. Pellegrini and A. Zalesski, On generators and representations of the sporadic simple groups, Comm. Algebra 42 (2014) 880--908.
J. Moori, Generating sets for F22 and its automorphism group, J. Algebra 159 (1993) 488--499.
J. Moori, Subgroups of 3-transposition groups generated by four 3-transpositions, Quaest. Math. 17 (1994) 483--94.
J. Moori, On the ranks of the Fischer group F22, Math. Japonica 43 (1996) 365--367.
R. Ree, A theorem on permutations, J. Combin. Theorey Ser. A 10 (1971) 174--175.
L.L. Scott, Matrices and cohomology, Ann. of Math. (2) 105 (1977), no. 3, 473--492.
J.Ward, Generation of Simple Groups by Conjugate Involutions, PhD Thesis, University of London, 2009.
R.A. Wilson, The Finite Simple Groups, Grad. Texts in Math. 251, Springer-Verlag, London, 2009.
A.J. Woldar, Representing M11; M12; M22 and M23 on surfaces of least genus, Comm. Algebra 18 (1990) 15--86.
I. Zisser, The covering numbers of the sporadic simple groups, Israel J. Math 67 (1989) 217--224.