\((p,q,r)\)-generations of the Conway group \(Co_1\) for odd \(p\) (Q2730851)

This paper is devoted to a detailed study of the different ways in which the largest Conway group \(Co_1\) can be generated by two elements, \(a\) and \(b\), say, with the orders of \(a\), \(b\) and \(ab\) being distinct odd primes \(p\), \(q\) and \(r\).NEWLINENEWLINENEWLINEFor each triple of conjugacy classes of elements of orders \(p\), \(q\), \(r\) respectively, it is determined whether or not there exists a generating triple \((a,b,ab)\) of elements in these conjugacy classes (with three exceptions, which the authors were apparently unable to resolve). The method is to use partial fusion of conjugacy classes from the maximal subgroups, to show that not all triples of a given type can generate proper subgroups.NEWLINENEWLINENEWLINERemarks: (1) The ATLAS list of maximal subgroups of \(Co_1\) has been corrected by the reviewer -- this paper gives the corrected list. (2) For the record, generating triples of the remaining three types do exist. They can easily be constructed explicitly using a computer algebra package such as MAGMA.