# Lyons group

Algebraic structure → Group theory Group theory :
Basic notions
Subgroup
Normal subgroup
Group action
Quotient group
(Semi-)direct product
Direct sum
Free product
Wreath product
Group homomorphisms
kernel
image
simple
finite
infinite
continuous
multiplicative
additive
cyclic
abelian
dihedral
nilpotent
solvable
Glossary of group theory
List of group theory topics :
Finite groups
Cyclic group Z$_n$
Symmetric group S$_n$
Alternating group A$_n$
Dihedral group D$_n$
Quaternion group Q
Cauchy's theorem
Lagrange's theorem
Sylow theorems
Hall's theorem
p-group
Elementary abelian group
Frobenius group
Schur multiplier
Classification of finite simple groups
cyclic
alternating
Lie type
sporadic :
Discrete groupsLattices
Integers ($\mathbb {Z}$)
Free group
Modular groups PSL(2, $\mathbb {Z}$)SL(2, $\mathbb {Z}$)
Arithmetic group
Lattice
Hyperbolic group :
Topological and Lie groups
Solenoid
Circle
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Euclidean E(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n)
G$_2$
F$_4$
E$_6$
E$_7$
E$_8$
Lorentz
Poincaré
Conformal
Diffeomorphism
Loop
Infinite dimensional Lie group O(∞)SU(∞)Sp(∞) :
Algebraic groups
Linear algebraic group
Reductive group
Abelian variety
Elliptic curve :
vte :
In the area of modern algebra known as group theory, the Lyons group Ly or Lyons-Sims group LyS is a sporadic simple group of order
    51,765,179,004,000,000
    = 2$^8$ · 3$^7$ · 5$^6$ · 7 · 11 · 31 · 37 · 67
    ≈ 5×10$^{16}$.

## History

Ly is one of the 26 sporadic groups and was discovered by Richard Lyons and Charles Sims in 1972-73. Lyons characterized 51765179004000000 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 A$_{11}$ of degree 11 by the cyclic group C$_2$. 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 group was discovered, it was noticed that a centralizer of one of its involutions was the perfect double cover of the alternating group A$_8$. This suggested considering the double covers of the other alternating groups A$_n$ 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 Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 67 are not supersingular primes, the Lyons group cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

## 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 G$_2$(5). There is also a slightly larger rank 5 permutation representation on 9606125 points with stabilizer 3.McL:2.

## Maximal subgroups

Wilson (1985) found the 9 conjugacy classes of maximal subgroups of Ly as follows:
Maximal subgroups of Ly :
| No. | Structure | Order | Index | Comments |
| 1 | G$_2$(5) | 5,859,000,000 = 2$^6$·3$^3$·5$^6$·7·31 | 8,835,156 = 2$^2$·3$^4$·11·37·67 | |
| 2 | 3$^·$McL:2 | 5,388,768,000 = 2$^8$·3$^7$·5$^3$·7·11 | 9,606,125 = 5$^3$·31·37·67 | normalizer of a subgroup of order 3 (class 3A) |
| 3 | 5$^{3·}$L$_3$(5) | 46,500,000 = 2$^5$·3·5$^6$·31 | 1,113,229,656 = 2$^3$·3$^6$·7·11·37·67 | |
| 4 | 2$^·$A$_{11}$ | 39,916,800 = 2$^8$·3$^4$·5$^2$·7·11 | 1,296,826,875 = 3$^3$·5$^4$·31·37·67 | centralizer of an involution |
| 5 | 5$^{1+4}$:4.S$_6$ | 9,000,000 = 2$^6$·3$^2$·5$^6$ | 5,751,686,556 = 2$^2$·3$^5$·7·11·31·37·67 | normalizer of a subgroup of order 5 (class 5A) |
| 6 | 3$^5$:(2 × M$_{11}$) | 3,849,120 = 2$^5$·3$^7$·5·11 | 13,448,575,000 = 2$^3$·5$^5$·7·31·37·67 | |
| 7 | 3$^{2+4}$:2.A$_5$.D$_8$ | 699,840 = 2$^6$·3$^7$·5 | 73,967,162,500 = 2$^2$·5$^5$·7·11·31·37·67 | |
| 8 | 67:22 | 1,474 = 2·11·67 | 35,118,846,000,000 = 2$^7$·3$^7$·5$^6$·7·31·37 | |
| 9 | 37:18 | 666 = 2·3$^2$·37 | 77,725,494,000,000 = 2$^7$·3$^5$·5$^6$·7·11·31·67 | |