Higman-Sims group
Encyclopedia
In the mathematical field of group theory
, the Higman–Sims group HS is a sporadic simple group found by of order
It is the simple subgroup of index
two in the group of automorphisms of the Higman–Sims graph. The Higman–Sims graph has 100 nodes, so the Higman–Sims group HS is a transitive group of permutations
of a 100 element set.
The Higman–Sims group was discovered in 1967, when Higman and Sims were attending a presentation by Marshall Hall
on the Hall–Janko group
. This is also a permutation group of 100 points, and the stabilizer of a point is a subgroup
with two other orbits of lengths 36 and 63. It occurred to them to look for a group of permutations of 100 points containing the Mathieu group
M22, which has permutation representations on 22 and 77 points. (The latter representation arises because the M22 Steiner system
has 77 blocks.) By putting together these two representations, they found HS, with a one-point stabilizer isomorphic to M22.
independently discovered the group as a doubly transitive permutation group acting on a certain 'geometry' on 176 points.
The Schur multiplier
has order 2, the outer automorphism group
has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group.
. Here, HS fixes a 2-3-3 triangle and a 22-dimensional sublattice. The Higman–Sims group thus becomes a subgroup of each of the Conway groups Co1, Co2 and Co3. If a conjugate of HS in Co1 fixes a particular point of type 3, this point is found in 276 triangles of type 2-2-3, which this copy of HS permutes in orbits of 176 and 100.
This also provides a 22 dimensional representation of HS, acting on a 22 dimensional lattice given by the orthogonal complement of a 2-3-3 triangle with one vertex ot the origin.
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 Higman–Sims group HS is a sporadic simple group found by 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....
- 29 · 32 · 53 · 7 · 11
- = 44352000.
- ≈ 4 · 107.
It is the simple subgroup of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
two in the group of automorphisms of the Higman–Sims graph. The Higman–Sims graph has 100 nodes, so the Higman–Sims group HS is a transitive group of permutations
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
of a 100 element set.
The Higman–Sims group was discovered in 1967, when Higman and Sims were attending a presentation by Marshall Hall
Marshall Hall (mathematician)
Marshall Hall, Jr. was an American mathematician who made contributions to group theory and combinatorics.- Career :...
on the Hall–Janko group
Hall–Janko group
In mathematics, the Hall-Janko group HJ, is a finite simple sporadic group of order 604800. It is also called the second Janko group J2, or the Hall-Janko-Wales group, since it was predicted by Janko and constructed by Hall and Wales...
. This is also a permutation group of 100 points, and the stabilizer of a point is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
with two other orbits of lengths 36 and 63. It occurred to them to look for a group of permutations of 100 points containing 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...
M22, which has permutation representations on 22 and 77 points. (The latter representation arises because the M22 Steiner system
Steiner system
250px|right|thumbnail|The [[Fano plane]] is an S Steiner triple system. The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line....
has 77 blocks.) By putting together these two representations, they found HS, with a one-point stabilizer isomorphic to M22.
independently discovered the group as a doubly transitive permutation group acting on a certain 'geometry' on 176 points.
The Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
has order 2, the outer automorphism group
Outer automorphism group
In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group.
Relationship with the Conway groups
showed how the Higman–Sims graph could be embedded in the Leech latticeLeech 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...
. Here, HS fixes a 2-3-3 triangle and a 22-dimensional sublattice. The Higman–Sims group thus becomes a subgroup of each of the Conway groups Co1, Co2 and Co3. If a conjugate of HS in Co1 fixes a particular point of type 3, this point is found in 276 triangles of type 2-2-3, which this copy of HS permutes in orbits of 176 and 100.
This also provides a 22 dimensional representation of HS, acting on a 22 dimensional lattice given by the orthogonal complement of a 2-3-3 triangle with one vertex ot the origin.
Maximal subgroups
showed that HS has 12 conjugacy classes of maximal subgroups.- M22, order 443520
- U3(5):2, order 252000 – one-point stabilizer in doubly transitive representation of degree 176
- U3(5):2 – conjugate to class above in HS:2
- PSL(3,4):2, order 40320
- S8, order 40320
- 24.S6, order 11520
- 43:PSL(3,2), order 10752
- M11, order 7920
- M11 – conjugate to class above in HS:2
- 4.24.S5, order 7680 – centralizer of involution moving 80 vertices of Higman–Sims graph
- 2 × A6.22, order 2880 – centralizer of involution moving all 100 vertices
- 5:4 × A5, order 1200