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Generating pairs for the Conway groups \(Co_2\) and \(Co_3\). (English) Zbl 0906.20020
This paper is devoted to a detailed study of the different ways in which the two groups of the title can be generated by two elements, \(a\) and \(b\), say. For symmetry we add a redundant generator \(c=b^{-1}a^{-1}\), and consider the triple of generators \((a,b,c)\) with \(abc=1\). Thus we can cyclically permute \(a,b,c\), and by replacing the generators by their inverse we can also achieve odd permutations of the cyclic groups they generate. We may therefore assume that \(o(a)\leq o(b)\leq o(c)\).
If these three orders are distinct primes \(p,q,r\), we speak of \((p,q,r)\)-generation, etc., and if \(a,b,c\) are specified further to lie in particular conjugacy classes \(pX,qY,rZ\), we speak of \((pX,qY,rZ)\)-generation. Section 2 is devoted to determining precisely which such triples of conjugacy classes arise for generating pairs as above, in the Conway group \(Co_2\). The group \(Co_3\) has been dealt with in an earlier paper [the authors, J. Algebra 188, No. 2, 516-530 (1997; Zbl 0874.20016)].
The second question which this paper answers, for each conjugacy class \(C\), is: can the group \(G\) be generated by an element in \(C\) and an element \(y\) in any nontrivial conjugacy class \(Y\) whatsoever? (That is, \(\forall x\in C\), \(\forall Y\), \(\exists y\in Y\): \(\langle x,y\rangle=G\).) This is relatively easy to answer, once the answer to the first question is known.

MSC:
20F05 Generators, relations, and presentations of groups
20D08 Simple groups: sporadic groups
Software:
GAP
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References:
[1] Conder M. D. E., Proc. Amer. Math. Soc. 116 pp 653– (1992)
[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson.Atlas of nite groups (Clarendon Press, 1985). · Zbl 0568.20001
[3] Curtis R. T., J. Algebra 27 pp 549– (1973)
[4] L. Di Martino and C. Tamburini. 2-generation of nite simple groups and some related topics. In Generators and relations in groups and geometries (Kluwer, 1991), pp. 195-233. · Zbl 0751.20027
[5] Finkelstein L., J. Algebra 25 pp 58– (1973)
[6] Ganief S., Comm. Algebra 23 pp 4427– (1995)
[7] Ganief S., Comm. Algebra 24 pp 809– (1996)
[8] Ganief S., J. Algebra 188 pp 531– (1997)
[9] Ganief S., J. Algebra 188 pp 516– (1997)
[10] Ganief S., Nova J. Math. Game Theory Algebra 6 pp 127– (1997)
[11] I. M. Isaacs.Character theory of nite groups (Dover, 1976). · Zbl 0337.20005
[12] Moori J., Nova J. Algebra Geom. 2 pp 277– (1993)
[13] Moori J., Comm. Algebra 22 pp 4597– (1994)
[14] Ree R., J. Combin. Theory Ser. 10 pp 174– (1971)
[15] M. SchoEnert et al. GAP 3.4 Manual groups, algorithms and programming (Lehrstuhl D fuEr Mathematik, RWTH, Aachen, 1992).
[16] L. L. Scott.Matrices and cohomology. Ann. of Math. (2) 105 (1977), 473-492. · Zbl 0399.20047
[17] Wilson R. A., J. Algebra 84 pp 107– (1983)
[18] Woldar A. J., Illinois J. Math. 33 pp 416– (1989)
[19] Woldar A. J., Comm. Algebra 22 pp 675– (1994)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.