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\((2,3,p)\)-generations for the Fischer group \(F_{22}\). (English) Zbl 0806.20018
The author studies ways of generating the smallest Fischer group \(\text{Fi}_{22}\), with elements \(x\) of order 2, \(y\) of order 3, with \(xy\) of prime order \(p\). Clearly \(p = 7\), 11 or 13, and the main result of this paper is that there are just seven possibilities for the conjugacy classes of \(x\), \(y\) and \(xy\), namely \((2C,3D,7A)\), \((2C,3D,11A/B)\) and \((2C,3C/D,13A/B)\).
The existence of the generating triples in the cases \(p = 7\) and \(p = 11\) was already known (and in particular, the fact that \(\text{Fi}_{22}\) is a Hurwitz group). The non-existence of other types of generating triples is proved by a combination of methods, involving Ree’s theorem on generators of transitive permutation groups, and the corresponding result for irreducible matrix groups due to Scott. The other main technique used is detailed analysis of structure constants in \(\text{Fi}_{22}\) and its subgroups, using some results on the maximal subgroups of \(\text{Fi}_{22}\).

MSC:
20D08 Simple groups: sporadic groups
20F05 Generators, relations, and presentations of groups
20E28 Maximal subgroups
20D30 Series and lattices of subgroups
Software:
Cayley
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