|Algebraic structure → Group theory
- 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
- = 51765179004000000
- ≈ 5 · 1016.
The Lyons group is one of the 6 sporadic simple groups called "pariah groups", i.e. those groups not found within the monster group (as the order of the monster group is not divisible by 37 or 67). As of 2012, the Lyons group appears to have no applications outside finite group theory.
In 1972, Richard Lyons characterized this number as the unique possible order of any finite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group A11 of degree 11 by the cyclic group C2. Sims (1973) proved the existence of such a group and its uniqueness up to isomorphism with a combination of permutation group theory and machine calculations.
When the McLaughlin sporadic simple group was discovered, it was noticed that a centralizer of one of its involutions was the perfect double cover of the alternating group A8. This suggested considering the double covers of the other alternating groups An as possible centralizers of involutions in simple groups. The cases n≤7 are ruled out by the Brauer-Suzuki theorem, the case n=8 leads to the McLaughlin group, the case n=9 was ruled out by Zvonimir Janko, Lyons himself ruled out the case n=10 and found the Lyons group for n=11, while the cases n≥12 were ruled out by J.G. Thompson and Ronald Solomon.
Meyer, Neutsch & Parker (1985) showed that the Lyons group has a modular representation of dimension 111 over the field of five elements, which is the smallest dimension of any faithful linear representation and is one of the easiest ways of calculating with it. It has also been given by several complicated presentations in terms of generators and relations, for instance those given by Sims (1973) or Gebhardt (2000).
The smallest faithful permutation representation is a rank 5 permutation representation on 8835156 points with stabilizer G2(5). There is also a slightly larger rank 5 permutation representation on 9606125 points with stabilizer 3.McL:2.
Wilson (1985) classified the maximal subgroups of the Lyons group, finding 9 classes, with structures G2(5), 3.McL:2, 53.PSL3(5), 2.A11, 51+4:4.S6, 35:(2 × M11), 32+4:2.A5.D8, 67:22, 37:18.
- Richard Lyons (1972,5) "Evidence for a new finite simple group", Journal of Algebra 20:540–569 and 34:188–189.
- Volker Gebhardt (2000) "Two short presentations for Lyons' sporadic simple group", Experimental Mathematics 9(3):333–8.
- Meyer, Werner; Neutsch, Wolfram; Parker, Richard (1985), "The minimal 5-representation of Lyons' sporadic group", Mathematische Annalen 272 (1): 29–39, doi:10.1007/BF01455926, ISSN 0025-5831, MR 794089
- Sims, Charles C. (1973), "The existence and uniqueness of Lyons' group", Finite groups '72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972), North-Holland Math. Studies 7, Amsterdam: North-Holland, pp. 138–141, MR 0354881
- Wilson, Robert A. (1985), "The maximal subgroups of the Lyons group", Mathematical Proceedings of the Cambridge Philosophical Society 97 (3): 433–436, doi:10.1017/S0305004100063003, ISSN 0305-0041, MR 778677