# zbMATH — the first resource for mathematics

$$(2,3,p)$$-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
Cayley
Full Text:
##### References:
 [1] Cannon, J. J. 1984.An introduction to the group theory language CAYLEY, 145–183. San Diego: Academic Press. [2] DOI: 10.1112/blms/20.3.235 · Zbl 0663.20003 [3] DOI: 10.1017/S0004972700001738 · Zbl 0546.20027 [4] DOI: 10.1090/S0002-9939-1992-1126192-2 [5] Conway J. H, Atlas of Finite Groups (1985) [6] Conway, J.H. 1973.A construction for the smallest Fischer group, 27–35. North-Holland: Gagen,Hale and Shult, eds. · Zbl 0267.20007 [7] Coxeter H. S. M, Generators and Relations for Discrete Groups (1980) [8] DOI: 10.1080/00927877808822270 · Zbl 0379.20017 [9] DOI: 10.1017/S0305004100067001 · Zbl 0622.20008 [10] DOI: 10.1112/jlms/s2-23.1.61 · Zbl 0443.20016 [11] Moori J., J. Algebra and Geometry 23 (1981) [12] DOI: 10.1016/0097-3165(71)90020-3 · Zbl 0221.05033 [13] Schonertetal M., Lehrstul D Fiir Mathematik 10 (1992) [14] DOI: 10.2307/1970920 · Zbl 0399.20047 [15] DOI: 10.1112/blms/2.3.319 · Zbl 0206.30804 [16] DOI: 10.1016/0095-8956(83)90009-6 · Zbl 0521.05027 [17] DOI: 10.1017/S0305004100061491 · Zbl 0551.20010 [18] Woldar A. J, Illinois Math, J 33 pp 416– (1989) [19] DOI: 10.1016/0021-8693(91)90115-O · Zbl 0736.20012 [20] DOI: 10.1080/00927879008823902 · Zbl 0701.20012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.