\(2\)-generations of the fourth Janko group \(J_4\) (Q1284112)

This paper is devoted to a detailed study of the different ways in which the group of the title can be generated by two elements, \(a\) and \(b\), say. We add a redundant generator \(c=b^{-1}a^{-1}\), and by symmetry we may assume that \(o(a)\leq o(b)\leq o(c)\). If these three orders are \(l\), \(m\), \(n\), we speak of \((l,m,n)\)-generation, etc., and if \(a\), \(b\), \(c\) are specified further to lie in particular conjugacy classes \(lX\), \(mY\), \(nZ\), we speak of \((lX,mY,nZ)\)-generation. Sections 2-4 are devoted to determining precisely which such triples of conjugacy classes arise for generating pairs as above, in the Janko group \(J_4\). The authors use the completeness of the list of maximal subgroups in an essential way. The second question is, for each conjugacy class \(C\): can the group \(J_4\) be generated by an element in \(C\) and an element \(y\) in any non-trivial conjugacy class \(Y\) whatsoever? This is known as \(C\)-complementary generation, and is relatively easy to determine, once the answer to the first question is known. Several earlier papers by the same authors, performing the same calculations in other sporadic simple groups, are referred to.