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 In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

: 2⁸3⁷5⁶711313767 : = 51765179004000000 : ≈ 5.

History

''Ly'' is one of the 26 sporadic groups and was discovered by Richard Lyons and

Charles Sims Charles Sims may refer to: * Charles Sims (painter) (1873–1928), British painter * Charles Sims (mathematician) (1938–2017), American mathematician * Charles Sims (aviator) (1899–1929), British World War I flying ace * Charles Sims (American ...

in 1972-73. Lyons characterized 51765179004000000 as the unique possible order of any finite simple group where the

centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...

of some

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

isomorphic In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...

to the nontrivial central extension of the

alternating group In mathematics, an alternating group is the 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 by or Basic prop ...

A₁₁ of degree 11 by the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

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 In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order : 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898,128,000 : ≈ 9. History and properties McL is one of the 26 spo ...

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 In mathematics, the Brauer–Suzuki theorem, proved by , , , states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a center of order 2. In particular, such ...

, the case ''n'' = 8 leads to the McLaughlin group, the case ''n'' = 9 was ruled out by

Zvonimir Janko Zvonimir Janko (26 July 1932 – 12 April 2022) was a Croatian mathematician who was the eponym of the Janko groups, sporadic simple groups in group theory. The first few sporadic simple groups were discovered by Émile Léonard Mathieu, which ...

, 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 In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...

and the

outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...

are both

trivial Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense. Latin Etymology The ancient Romans used the word ''triviae'' to describe where one road split or forked ...

. Since 37 and 67 are not

supersingular In mathematics, a supersingular variety is (usually) a smooth projective variety in nonzero characteristic such that for all ''n'' the slopes of the Newton polygon of the ''n''th crystalline cohomology are all ''n''/2 . For special classes o ...

primes, the Lyons group cannot be a

subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...

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, having order 246320597611213317192329314147 ...

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

pariahs Pariah may refer to: * A member of the Paraiyar caste in the Indian state of Tamil Nadu * Pariah state, a country whose behavior does not conform to norms * Outcast (person) Science and mathematics * Pariah dog, a type of semi-feral dog * ''Pa ...

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 In mathematics, the term permutation representation of a (typically finite) group G can refer to either of two closely related notions: a representation of G as a group of permutations, or as a group of permutation matrices. The term also refers t ...

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.

Maximal subgroups

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

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. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...

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