×

On the ranks of Conway group \(Co_1\). (English) Zbl 1083.20014

Summary: Let \(G\) be a finite group and \(X\) a conjugacy class of \(G\). We denote \(\text{rank}(G:X)\) to be the minimum number of elements of \(X\) generating \(G\). In the present paper we investigate the ranks of the Conway group \(Co_1\). Computations are carried out with the aid of computer algebra system GAP [The GAP Group, GAP – Groups, Algorithms and Programming, http://www-gap.dcs.st-and.ac.uk/gap].

MSC:

20D08 Simple groups: sporadic groups
20F05 Generators, relations, and presentations of groups
20E45 Conjugacy classes for groups
20C40 Computational methods (representations of groups) (MSC2010)

Software:

GAP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. Ali and M. A. F. Ibrahim, On the ranks of \(HS\) and \(McL\), Utilitas Mathematica, (2005). (To appear).
[2] F. Ali and M. A. F. Ibrahim, On the ranks of \(Co_2\) and \(Co_3\). (Submitted). · Zbl 1082.20003
[3] M. Aschbacher, Sporadic Groups , Cambridge Univ. Press, London-New York, 1994. · Zbl 0804.20011
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of finite groups , Oxford Univ. Press, Eynsham, 1985. · Zbl 0568.20001
[5] M. D. E. Conder, R. A. Wilson and A. J. Woldar, The symmetric genus of sporadic groups, Proc. Amer. Math. Soc. 116 (1992), no. 3, 653-663. · Zbl 0836.20014 · doi:10.2307/2159431
[6] M. R. Darafsheh and A. R. Ashrafi, \((2,p,q)\)-generations of the Conway group \(\mathrm{Co}_1\), Kumamoto J. Math. 13 (2000), 1-20. · Zbl 0949.20012
[7] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, \((p,q,r)\)-generations of the Conway group \(\mathrm{Co}_1\) for odd \(p\), Kumamoto J. Math. 14 (2001), 1-20. · Zbl 0980.20007
[8] M. R. Darafsheh, A. R. Ashrafi and G. A. Moghani, \(nX\)-complementary generations of the sporadic group \(\mathrm{Co}_1\), Acta Math. Vietnam. 29 (2004), no. 1, 57-75. · Zbl 1074.20010
[9] S. Ganief and J. Moori, Generating pairs for the Conway groups \(\mathrm{Co}_2\) and \(\mathrm{Co}_3\), J. Group Theory 1 (1998), no. 3, 237-256. · Zbl 0906.20020 · doi:10.1515/jgth.1998.016
[10] J. I. Hall and L. H. Soicher, Presentations of some \(3\)-transposition groups, Comm. Algebra 23 (1995), no. 7, 2517-2559. · Zbl 0830.20052 · doi:10.1080/00927879508825358
[11] I. M. Isaacs, Character theory of finite groups , Corrected reprint of the 1976 original [Academic Press, New York], Dover, New York, 1994.
[12] J. Moori, Generating sets for \(F_{22}\) and its automorphism group, J. Algebra 159 (1993), no. 2, 488-499. · Zbl 0799.20017 · doi:10.1006/jabr.1993.1170
[13] J. Moori, Subgroups of \(3\)-transposition groups generated by four \(3\)-transpositions, Quaestiones Math. 17 (1994), no. 1, 83-94. · Zbl 0822.20013 · doi:10.1080/16073606.1994.9632219
[14] J. Moori, On the ranks of the Fischer group \(F_{22}\), Math. Japon. 43 (1996), no. 2, 365-367. · Zbl 0849.20012
[15] J. Moori, On the ranks of Janko groups \(J_1\), \(J_2\) and \(J_3\), in 41st annual congress of South African Mathematical Society , RAU, Auckland Park, (1998). (Private Communication).
[16] The GAP Group, \(\GAP\) - Groups, Algorithms and programming, version 4.3, Aachen, St Andrews, 2003. (http://www-gapdcsst-andacuk/ gap)
[17] L. L. Scott, Matrices and cohomology, Ann. of Math. (2) 105 (1977), no. 3, 473-492. · Zbl 0399.20047 · doi:10.2307/1970920
[18] R. A. Wilson, The maximal subgroups of Conway’s group \(\mathrm{Co}_{1}\), J. Algebra 85 (1983), no. 1, 144-165. · Zbl 0525.20009 · doi:10.1016/0021-8693(83)90122-9
[19] A. J. Woldar, Representing \(M_{11}\), \(M_{12}\), \(M_{22}\) and \(M_{23}\) on surfaces of least genus, Comm. Algebra 18 (1990), no. 1, 15-86. · Zbl 0701.20012 · doi:10.1080/00927879008823902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.