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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.

20F05 Generators, relations, and presentations of groups
20D08 Simple groups: sporadic groups
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