an:00636115
Zbl 0806.20018
Moori, Jamshid
\((2,3,p)\)-generations for the Fischer group \(F_{22}\)
EN
Commun. Algebra 22, No. 11, 4597-4610 (1994).
0092-7872 1532-4125
1994
j
20D08 20F05 20E28 20D30
Fischer group \(\text{Fi}_{22}\); conjugacy classes; generating triples; Hurwitz group; generators; maximal subgroups
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 \(\text{Fi}_{22}\).
R.Wilson (Birmingham)