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In the area of modern algebra known as group theory, the Lyons groupLy or Lyons-Sims groupLyS is a sporadic simple group of order
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 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 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 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.
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.
Wilson (1985) found the 9 conjugacy classes of maximal subgroups of Ly as follows:
In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
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
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
= 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
≈ 8×1053.
In the mathematical classification of finite simple groups, there are 26 or 27 groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups.
In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by (Conway 1968, 1969).
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by Mathieu. They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered.
In the area of modern algebra known as group theory, the Suzuki groupSuz or Sz is a sporadic simple group of order
In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order
In group theory, the Tits group2F4(2)′, named for Jacques Tits (French:[tits]), is a finite simple group of order
In the area of modern algebra known as group theory, the Held groupHe is a sporadic simple group of order
In the area of abstract algebra known as group theory, the O'Nan groupO'N or O'Nan–Sims group is a sporadic simple group of order
In the area of modern algebra known as group theory, the Rudvalis groupRu is a sporadic simple group of order
In the area of modern algebra known as group theory, the Janko groupJ1 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Janko groupJ2 or the Hall-Janko groupHJ is a sporadic simple group of order
In the area of modern algebra known as group theory, the Mathieu groupM22 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Mathieu groupM24 is a sporadic simple group of order
In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order
In the area of modern algebra known as group theory, the Conway groupCo2 is a sporadic simple group of order
In the area of modern algebra known as group theory, the Conway group is a sporadic simple group of order
In the area of modern algebra known as group theory, the Conway groupCo1 is a sporadic simple group of order
