Definitions

Lyons group

In the mathematical field of group theory, the Lyons group Ly (whose existence was suggested by Richard Lyons in 1970), is a sporadic simple group of order

28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
= 51765179004000000
≈ 5 · 10 16 .

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. The existence of such a group and its uniqueness up to isomorphism was proved with a combination of permutation group theory and clever machine calculations by C. C. Sims. The group is also known as the Lyons-Sims group LyS.

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.

The Lyons group can be described more concretely in terms of a modular representation of dimension 111 over the field of five elements, or in terms of generators and relations, for instance those given by Gebhardt (2000).

Ly is one of the 6 sporadic simple groups called the pariahs, those which are not found within the monster group (as the order of the monster group is not divisible by 37 or 67).

References

• R. Lyons, Evidence for a new finite simple group, J. Algebra 20 (1972) 540-569 and 34 (1975) 188-189.
• Volker Gebhardt, Two short presentations for Lyons' sporadic simple group, Experimental Mathematics, 9 (2000) no. 3, 333-338.