# Lyons group

In mathematical group theory, the Lyons group Ly or Lyons-Sims group LyS is a sporadic simple group of order

28 · 37 · 56 ·· 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.

## History

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.

## Representations

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.

## Subgroups

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.