In the area of modern algebra known as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, the Lyons group ''Ly'' or Lyons-Sims group ''LyS'' is a sporadic simple group of order : 51,765,179,004,000,000 : = 2⁸3⁷5⁶711313767 : ≈ 5.

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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the nontrivial central extension of the

alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...

A₁₁ of degree 11 by the

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

C₂. 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

''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 Ronald "Ron" Mark Solomon (b. 15 December 1948Mathematicians who classified finite groups ...

. 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 In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order : : = 2463205976112133171923293 ...

. Thus it is one of the 6 sporadic groups called the pariahs.

Representations

showed that the Lyons group has a

modular representation Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as h ...

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 or . 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. The degrees of irreducible representations of the Lyons group are 1, 2480, 2480, 45694, 48174, ... .

Maximal subgroups

found the 9 conjugacy classes of maximal subgroups of ''Ly'' as follows:

References

* Richard Lyons (1972,5) "Evidence for a new finite simple group",

Journal of Algebra ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, te ...

20:540–569 and 34:188–189. * * * * {{Citation , last1=Wilson , first1=Robert A. , title=The maximal subgroups of the Lyons group , doi= 10.1017/S0305004100063003 , mr=778677 , year=1985 , journal= Mathematical Proceedings of the Cambridge Philosophical Society , issn=0305-0041 , volume=97 , issue=3 , pages=433–436, s2cid=119577612

External links

MathWorld: Lyons group

Atlas of Finite Group Representations: Lyons group
Sporadic groups