Higman–Sims group

In the mathematical field of group theory, the Higman–Sims group "HS" (named after Donald G. Higman and Charles C. Sims) is a finite group of order

: 2⁹ · 3² · 5³ · 7 · 11: = 44352000.: ≈ 4 · 10⁷.

It is a "simple group", meaning it does not have any normal subgroups except for the subgroup consisting only of the identity element, and "HS" itself.

The Higman–Sims group is one of the 26 sporadic groups. It can be characterized as 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.

"Higman" may also refer to the mathematician Graham Higman of the University of Oxford who independently discovered the group as the automorphism group of a certain 'geometry' on 176 points. Consequently, "HS" has a doubly-transitive permutation representation on 176 points.

Relationship with the Conway Groups

In his now classic 1968 paper, John Horton Conway showed how the Higman-Sims graph could be embedded in the Leech lattice. Here, "HS" fixes a 2-3-3 triangle and a 22-dimensional sublattice. The 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 provides an explicit way of approaching a low dimensional representation of the group, and with it, a straightforward means of carrying out computations inside the group.

HS is part of the second generation of sporadic groups, i. e. one of the 7 sporadic simple groups found in Co₁ that are not Mathieu groups.

Maximal subgroups

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

References

* | year=1968 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=61 | pages=398–400
* | year=1996 | volume=163
* | year=1976 | journal=Mathematics Magazine | issn=0025-570X | volume=49 | issue=4 | pages=163–180
* | year=1998
* | year=1968 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=105 | pages=110–113
* | year=1969 | journal=Illinois Journal of Mathematics | issn=0019-2082 | volume=13 | pages=74–80
* | year=1971 | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=77 | pages=535–539

External links

* [http://mathworld.wolfram.com/HigmanSimsGroup.html MathWorld: Higman-Sims Group]
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/HS/ Atlas of Finite Group Representations: Higman-Sims group]