Lyons group
Encyclopedia
In the mathematical field of group theory
, the Lyons group Ly (whose existence was suggested by Richard Lyons in 1970), is a sporadic
simple group
of order
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. The existence of such a group and its uniqueness up to isomorphism was proved with a combination of permutation group theory and clever machine calculations by C. C. Sims. The group is also known as the Lyons-Sims group LyS.
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.
The Lyons group can be described more concretely in terms of a modular representation of dimension 111 over the field of five elements, or in terms of generators and relations, for instance those given by Gebhardt (2000).
Ly is one of the 6 sporadic simple groups called the pariahs
, those which are not found within the monster group
(as the order of the monster group is not divisible by 37 or 67).
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the Lyons group Ly (whose existence was suggested by Richard Lyons in 1970), is a sporadic
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
of order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
- 28 · 37 · 56 · 7 · 11 · 31 · 37 · 67
- = 51765179004000000
- ≈ 5 · 10 16 .
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
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
C2. The existence of such a group and its uniqueness up to isomorphism was proved with a combination of permutation group theory and clever machine calculations by C. C. Sims. The group is also known as the Lyons-Sims group LyS.
When the McLaughlin sporadic simple group
McLaughlin group (mathematics)
In the mathematical discipline known as group theory, the McLaughlin group McL is a sporadic simple group of order 27 · 36 · 53· 7 · 11 = 898,128,000, discovered by as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 =1+112+162 vertices...
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
Zvonimir Janko
Zvonimir Janko is a Croatian mathematician who is the eponym of the Janko groups, sporadic simple groups in group theory.Janko was born in Bjelovar, Croatia. He studied at the University of Zagreb where he received Ph.D. in 1960. He then taught physics at a high school in Široki Brijeg in Bosnia...
, 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 Lyons group can be described more concretely in terms of a modular representation of dimension 111 over the field of five elements, or in terms of generators and relations, for instance those given by Gebhardt (2000).
Ly is one of the 6 sporadic simple groups called the pariahs
Pariah group
In mathematical group theory, the term pariah was introduced by to refer to the six sporadic simple groups that are not subquotients of the monster simple group.These groups are:*Three of the Janko groups: J1, J3, and J4.*The Lyons group...
, those which are not found within the monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
(as the order of the monster group is not divisible by 37 or 67).