Conway group
Encyclopedia
In mathematics
, the Conway groups Co1, Co2, and Co3 are three sporadic group
s discovered by John Horton Conway
.
The largest of the Conway groups, Co1, of order
is obtained as the quotient of Co0 (group of automorphisms of the Leech lattice
Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II25,1
. The groups Co2 (of order 42,305,421,312,000) and Co3 (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of type 2 and a vector of type 3 respectively. (The type of a vector is half of its square norm, v·v.) As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co1.
about 1964 investigated close packings of spheres in Euclidean spaces of large dimension. One of Leech's discoveries was a lattice packing in 24-space, based on what came to be called the Leech lattice Λ. He wondered whether his lattice's symmetry group contained an interesting simple group, but felt he needed the help of someone better acquainted with group theory. He had to do much asking around because the mathematicians were pre-occupied with agendas of their own. John Conway agreed to look at the problem. John G. Thompson
said he would be interested if he were given the order of the group. Conway expected to spend months or years on the problem, but found results in just a few sessions.
Two of these (subgroups of Co2 and Co3) can be defined as pointwise stabilizers of triangles with vertices, of sum zero, of types 2 and 3. A 2-2-3 triangle is fixed by the McLaughlin group
McL (order 898,128,000). A 2-3-3 triangle is fixed by the Higman-Sims group
(order 44,352,000).
Two other sporadic groups can be defined as stabilizers of structures on the Leech lattice. Identifying R24 with C12 and Λ with
the resulting automorphism group, i.e., the group of Leech lattice automorphisms preserving the complex structure, when divided by the 6-element group of complex scalar matrices, gives the Suzuki group
Suz (of order 448,345,497,600). Suz is the only sporadic proper subquotient
of Co1 that retains 13 as a prime factor. This group was discovered by Michio Suzuki
in 1968.
A similar construction gives the Hall-Janko group J2 (of order 604,800) as the quotient of the group of quaternion
ic automorphisms of Λ by the group ±1 of scalars.
The 7 simple groups described above comprise what Robert Griess calls the second generation of the Happy Family, which consists of the 20 sporadic simple groups found within the Monster group
. Several of the 7 groups contain at least some of the 5 Mathieu groups, which comprise the first generation.
There was a conference on group theory held May 2–4, 1968, at Harvard University. Richard Brauer
and Chih-Han Sah later published a book of its proceedings. It included important lectures on four groups of the second generation, but was a little too early to include the Conway groups. It has on the other hand been observed that if Conway had started a few years earlier, he could have discovered all 7 groups. Conway unified 4 seemingly rather unrelated groups into one larger group.
, whose automorphism group is the Mathieu group
M24. Let E be a multiplicative representation of this code, a group of diagonal 24-by-24 matrices whose diagonal elements equal 1 or -1. E is an abelian group of type 212. Define N as the holomorph
E:M24. Conway found that N is a maximal subgroup of Co0 and contains 2-Sylow subgroups of Co0. He used N to deduce the order of Co0.
The negative of the identity is in E and commutes with every 24-by-24 matrix. Then Co1 has a maximal subgroup with structure 211:M24.
The matrices of N have components that are integers. Since N is maximal in Co0, N is the group of all integral matrices in Co0.
The maximal subgroups of Co1 are as follows.
Co1 contains non-abelian simple groups of some 35 isomorphism types, as subgroups or as quotients of subgroups.
Co3 has a doubly transitive permutation representation on 276 type 2-2-3 triangles containing a fixed type 3 point.
It was fruitful to investigate the normalizers of smaller subgroups of the form 2.An. Several other maximal subgroups of Co0 are found in this way. Moreover, two sporadic groups appear in the resulting chain.
There is a subgroup 2.A8 x S4, but it is not maximal in Co0. Next there is the subgroup (2.A7 x PSL2(7)):2, whose order is divisible by 49. This group and the rest of the chain are maximal in Co0. Next comes (2.A6 x SU3(3)):2. The unitary group SU3(3) (order 6048) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is (2.A5 o 2.HJ):2. The aforementioned graph expands to the Hall-Janko graph
, with 100 vertices. The Hall-Janko group HJ makes its appearance here. Next comes (2.A4 o 2.G2(4)):2. G2(4) is an exceptional group of Lie type
. Its order is divisible by 13, fairly rare among subgroups of the Conway groups.
The chain ends with 6.Suz:2 (Suz=Suzuki group), which, as mentioned above, respects a complex representaion of the Leech Lattice.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Conway groups Co1, Co2, and Co3 are three sporadic group
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...
s discovered by John Horton Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
.
The largest of the Conway groups, Co1, of order
- 4,157,776,806,543,360,000,
is obtained as the quotient of Co0 (group of automorphisms of the Leech lattice
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II25,1
II25,1
In mathematics, II25,1 is the even 26-dimensional Lorentzian unimodular lattice. It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector...
. The groups Co2 (of order 42,305,421,312,000) and Co3 (of order 495,766,656,000) consist of the automorphisms of Λ fixing a lattice vector of type 2 and a vector of type 3 respectively. (The type of a vector is half of its square norm, v·v.) As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co1.
History
Thomas Thompson relates how John LeechJohn Leech (mathematician)
John Leech was a British mathematician working in number theory, geometry and combinatorial group theory. He is best known for his discovery of the Leech lattice in 1965. He also discovered Ta in 1957...
about 1964 investigated close packings of spheres in Euclidean spaces of large dimension. One of Leech's discoveries was a lattice packing in 24-space, based on what came to be called the Leech lattice Λ. He wondered whether his lattice's symmetry group contained an interesting simple group, but felt he needed the help of someone better acquainted with group theory. He had to do much asking around because the mathematicians were pre-occupied with agendas of their own. John Conway agreed to look at the problem. John G. Thompson
John G. Thompson
John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize....
said he would be interested if he were given the order of the group. Conway expected to spend months or years on the problem, but found results in just a few sessions.
Other sporadic groups
Conway and Thompson found that 4 recently discovered sporadic simple groups were isomorphic to subgroups or quotients of subgroups of Co1.Two of these (subgroups of Co2 and Co3) can be defined as pointwise stabilizers of triangles with vertices, of sum zero, of types 2 and 3. A 2-2-3 triangle is fixed by the McLaughlin 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...
McL (order 898,128,000). A 2-3-3 triangle is fixed by the Higman-Sims group
Higman-Sims group
In the mathematical field of group theory, the Higman–Sims group HS is a sporadic simple group found by of orderThe Higman–Sims group was discovered in 1967, when Higman and Sims were attending a presentation by Marshall Hall on the Hall–Janko group...
(order 44,352,000).
Two other sporadic groups can be defined as stabilizers of structures on the Leech lattice. Identifying R24 with C12 and Λ with
- Z[e2πi/3]12,
the resulting automorphism group, i.e., the group of Leech lattice automorphisms preserving the complex structure, when divided by the 6-element group of complex scalar matrices, gives the Suzuki group
Suzuki sporadic group
In mathematical group theory, the Suzuki group Suz or Sz is a sporadic simple group of order 213 · 37 · 52· 7 · 11 · 13 = 448345497600 discovered by . It is not related to the Suzuki groups of Lie type....
Suz (of order 448,345,497,600). Suz is the only sporadic proper subquotient
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 group theory, where they are also known as sections, and in abelian categories....
of Co1 that retains 13 as a prime factor. This group was discovered by Michio Suzuki
Michio Suzuki
was a Japanese mathematician who studied group theory.-Biography:He was a Professor at the University of Illinois at Urbana-Champaign from 1953 to his death. He also had visiting positions at the University of Chicago , the Institute for Advanced Study , the University of Tokyo , and the...
in 1968.
A similar construction gives the Hall-Janko group J2 (of order 604,800) as the quotient of the group of quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
ic automorphisms of Λ by the group ±1 of scalars.
The 7 simple groups described above comprise what Robert Griess calls the second generation of the Happy Family, which consists of the 20 sporadic simple groups 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:...
. Several of the 7 groups contain at least some of the 5 Mathieu groups, which comprise the first generation.
There was a conference on group theory held May 2–4, 1968, at Harvard University. Richard Brauer
Richard Brauer
Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory...
and Chih-Han Sah later published a book of its proceedings. It included important lectures on four groups of the second generation, but was a little too early to include the Conway groups. It has on the other hand been observed that if Conway had started a few years earlier, he could have discovered all 7 groups. Conway unified 4 seemingly rather unrelated groups into one larger group.
An important maximal subgroup of Co0
Conway started his investigation with a subgroup called N. The Leech lattice is defined by use of the binary Golay codeBinary Golay code
In mathematics and electronics engineering, a binary Golay code is a type of error-correcting code used in digital communications. The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics....
, whose automorphism group is the Mathieu group
Mathieu group
In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
M24. Let E be a multiplicative representation of this code, a group of diagonal 24-by-24 matrices whose diagonal elements equal 1 or -1. E is an abelian group of type 212. Define N as the holomorph
Holomorph (mathematics)
In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group which simultaneously contains the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in...
E:M24. Conway found that N is a maximal subgroup of Co0 and contains 2-Sylow subgroups of Co0. He used N to deduce the order of Co0.
The negative of the identity is in E and commutes with every 24-by-24 matrix. Then Co1 has a maximal subgroup with structure 211:M24.
The matrices of N have components that are integers. Since N is maximal in Co0, N is the group of all integral matrices in Co0.
Maximal subgroups of Co1
Co1 has 22 conjugacy classes of maximal subgroups.The maximal subgroups of Co1 are as follows.
- Co2
- 3.Suz:2 (order divisible by 13)
- 211:M24
- Co3
- 21+8.O8+(2)
- U6(2):S3
- (A4 × G2(4)):2 (order divisible by 13)
- 22+12:(A8 × S3)
- 24+12.(S3 × 3.S6)
- 32.U4(3).D8
- 36:2.M12 (holomorph of ternary Golay codeTernary Golay codeThere are two closely related error-correcting codes known as ternary Golay codes. The code generally known simply as the ternary Golay code is a perfect [11, 6, 5] ternary linear code; the extended ternary Golay code is a [12, 6, 6] linear code obtained by adding a zero-sum check digit to the...
) - (A5 × J2):2
- 31+4:2.Sp4(3).2
- (A6 × U3(3)).2
- 33+4:2.(S4 × S4)
- A9 × S3
- (A7 × L2(7)):2 (order divisible by 49)
- (D10 × (A5 × A5).2).2
- 51+2:GL2(5) (contains Sylow 5-subgroups of Co1)
- 53:(4 × A5).2 (contains Sylow 5-subgroups of Co1)
- 72:(3 × 2.S4) (order divisible by 49)
- 52:2A5
Co1 contains non-abelian simple groups of some 35 isomorphism types, as subgroups or as quotients of subgroups.
Maximal subgroups of Co2
There are 11 conjugacy classes of maximal subgroups.- U6(2):2
- 210:M22:2
- McL (fixing 2-2-3 triangle)
- 21+8:Sp6(2)
- HS:2 (can transpose type 3 vertices of conserved 2-3-3 triangle)
- (24 × 21+6).A8
- U4(3):D8
- 24+10.(S5 × S3)
- M23
- 31+4.21+4.S5
- 51+2:4S4
Maximal subgroups of Co3
There are 14 conjugacy classes of maximal subgroups.Co3 has a doubly transitive permutation representation on 276 type 2-2-3 triangles containing a fixed type 3 point.
- McL:2 - can transpose type 2 points of conserved 2-2-3 triangle
- HS - fixes 2-3-3 triangle
- U4(3).22
- M23
- 35:(2 × M11)
- 2.Sp6(2) - centralizer of involution class 2A, which moves 240 of the 276 type 2-2-3 triangles
- U3(5):S3
- 31+4:4S6
- 24.A8
- PSL(3,4):(2 × S3)
- 2 × M12 - centralizer of involution class 2B, which moves 264 of the 276 type 2-2-3 triangles
- [210.33]
- S3 × PSL(2,8):3
- A4 × S5
A chain of product groups
Co0 (as well as its quotient Co1) has 4 conjugacy classes of elements of order 3. One of these commutes with a double cover of the alternating group A9. In fact the normalizer of that 3-element has the form 2.A9 x S3. This maximal subgroup reveals interesting features not found in the Mathieu groups. It has a simple subgroup of order 504, containing an element of order 9.It was fruitful to investigate the normalizers of smaller subgroups of the form 2.An. Several other maximal subgroups of Co0 are found in this way. Moreover, two sporadic groups appear in the resulting chain.
There is a subgroup 2.A8 x S4, but it is not maximal in Co0. Next there is the subgroup (2.A7 x PSL2(7)):2, whose order is divisible by 49. This group and the rest of the chain are maximal in Co0. Next comes (2.A6 x SU3(3)):2. The unitary group SU3(3) (order 6048) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is (2.A5 o 2.HJ):2. The aforementioned graph expands to the Hall-Janko graph
Hall-Janko graph
In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-regular undirected graph with 100 vertices and 1800 edges....
, with 100 vertices. The Hall-Janko group HJ makes its appearance here. Next comes (2.A4 o 2.G2(4)):2. G2(4) is an exceptional group of Lie type
Group of Lie type
In mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
. Its order is divisible by 13, fairly rare among subgroups of the Conway groups.
The chain ends with 6.Suz:2 (Suz=Suzuki group), which, as mentioned above, respects a complex representaion of the Leech Lattice.