Rank 3 permutation group
In mathematical finite group theory, a rank 3 permutation group acts transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by Higman (1964, 1971). Several of the sporadic simple groups were discovered as rank 3 permutation groups.
The primitive rank 3 permutation groups are all in one of the following classes:
- Cameron (1981) classified the ones such that where the socle T of T0 is simple, and T0 is a 2-transitive group of degree √n.
- Liebeck (1987) classified the ones with a regular elementary abelian normal subgroup
- Bannai (1971–72) classified the ones whose socle is a simple alternating group
- Kantor & Liebler (1982) classified the ones whose socle is a simple classical group
- Liebeck & Saxl (1986) classified the ones whose socle is a simple exceptional or sporadic group.
If G is any 4-transitive group acting on a set S, then its action on pairs of elements of S is a rank 3 permutation group. In particular most of the alternating groups, symmetric groups, and Mathieu groups have 4-transitive actions, and so can be made into rank 3 permutation groups.
The projective general linear group acting on lines in a projective space of dimension at least 3 is a rank-3 permutation group.
Several 3-transposition groups are rank-3 permutation groups (in the action on transpositions).
It is common for the point-stabilizer of a rank-3 permutation group acting on one of the orbits to be a rank-3 permutation group. This gives several "chains" of rank-3 permutation groups, such as the Suzuki chain and the chain ending with the Fischer groups.
Some unusual rank-3 permutation groups (many from (Liebeck & Saxl 1986)) are listed below.
|A6 = L2(9) = Sp4(2)' = M10'||S4||15 = 1+6+8||Pairs of points, or sets of 3 blocks of 2, in the 6-point permutation representation; two classes|
|A9||L2(8):3||120 = 1+56+63||Projective line P1(8); two classes|
|A10||(A5×A5):4||126 = 1+25+100||Sets of 2 blocks of 5 in the natural 10-point permutation representation|
|L2(8)||7:2 = Dih(7)||36 = 1+14+21||Pairs of points in P1(8)|
|L3(4)||A6||56 = 1+10+45||Hyperovals in P2(4); three classes|
|L4(3)||PSp4(3):2||117 = 1+36+80||Symplectic polarities of P3(3); two classes|
|G2(2)' = U3(3)||PSL3(2)||36 = 1+14+21||Suzuki chain|
|U3(5)||A7||50 = 1+7+42||The action on the vertices of the Hoffman-Singleton graph; three classes|
|U4(3)||L3(4)||162 = 1+56+105||Two classes|
|Sp6(2)||G2(2) = U3(3):2||120 = 1+56+63||The Chevalley group of type G2 acting on the octonion algebra over GF(2)|
|Ω7(3)||G2(3)||1080 = 1+351+728||The Chevalley group of type G2 acting on the imaginary octonions of the octonion algebra over GF(3); two classes|
|U6(2)||U4(3):22||1408 = 1+567+840||The point stabilizer is the image of the linear representation resulting from "bringing down" the complex representation of Mitchell's group (a complex reflection group) modulo 2; three classes|
|M11||M9:2 = 32:SD16||55 = 1+18+36||Pairs of points in the 11-point permutation representation|
|M12||M10:2 = A6.22 = PΓL(2,9)||66 = 1+20+45||Pairs of points, or pairs of complementary blocks of S(5,6,12), in the 12-point permutation representation; two classes|
|M22||24:A6||77 = 1+16+60||Blocks of S(3,6,22)|
|J2||PSU3(3)||100 = 1+36+63||Suzuki chain; the action on the vertices of the Hall-Janko graph|
|Higman-Sims group HS||M22||100 = 1+22+77||The action on the vertices of the Higman-Sims graph|
|M22||A7||176 = 1+70+105||Two classes|
|M23||M21:2 = L3(4):22 = PΣL(3,4)||253 = 1+42+210||Pairs of points in the 23-point permutation representation|
|M23||24:A7||253 = 1+112+140||Blocks of S(4,7,23)|
|McLaughlin group McL||U4(3)||275 = 1+112+162||The action on the vertices of the McLaughlin graph|
|M24||M22:2||276 = 1+44+231||Pairs of points in the 24-point permutation representation|
|G2(3)||U3(3):2||351 = 1+126+244||Two classes|
|G2(4)||J2||416 = 1+100+315||Suzuki chain|
|M24||M12:2||1288 = 1+495+792||Pairs of complementary dodecads in the 24-point permutation representation|
|Suzuki group Suz||G2(4)||1782 = 1+416+1365||Suzuki chain|
|G2(4)||U3(4):2||2016 = 1+975+1040|
|Co2||PSU6(2):2||2300 = 1+1891+1408|
|Rudvalis group Ru||²F₄(2)||4060 = 1+1755+2304|
|Fi22||2.PSU6(2)||3510 = 1+693+2816||3-transpositions|
|Fi22||Ω7(3)||14080 = 1+3159+10920||Two classes|
|Fi23||2.Fi22||31671 = 1+3510+28160||3-transpositions|
|G2(8).3||SU3(8).6||130816 = 1+32319+98496|
|Fi23||PΩ8+(3).S3||137632 = 1+28431+109200|
|Fi24'||Fi23||306936 = 1+31671+275264||3-transpositions|
- The three orbits are: the fixed pair itself; those pairs having one element in common with the fixed pair; and those pairs having no element in common with the fixed pair.
- Bannai, Eiichi (1971–72), "Maximal subgroups of low rank of finite symmetric and alternating groups", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 18: 475–486, ISSN 0040-8980, MR 0357559
- Brouwer, A. E.; Cohen, A. M.; Neumaier, Arnold (1989), Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 18, Berlin, New York: Springer-Verlag, ISBN 978-3-540-50619-5, MR 1002568
- Cameron, Peter J. (1981), "Finite permutation groups and finite simple groups", The Bulletin of the London Mathematical Society 13 (1): 1–22, doi:10.1112/blms/13.1.1, ISSN 0024-6093, MR 599634
- Higman, Donald G. (1964), "Finite permutation groups of rank 3", Mathematische Zeitschrift 86: 145–156, doi:10.1007/BF01111335, ISSN 0025-5874, MR 0186724
- Higman, Donald G. (1971), "A survey of some questions and results about rank 3 permutation groups", Actes du Congrès International des Mathématiciens (Nice, 1970) 1, Gauthier-Villars, pp. 361–365, MR 0427435
- Kantor, William M.; Liebler, Robert A. (1982), "The rank 3 permutation representations of the finite classical groups", Transactions of the American Mathematical Society 271 (1): 1–71, doi:10.2307/1998750, ISSN 0002-9947, MR 648077
- Liebeck, Martin W. (1987), "The affine permutation groups of rank three", Proceedings of the London Mathematical Society. Third Series 54 (3): 477–516, doi:10.1112/plms/s3-54.3.477, ISSN 0024-6115, MR 879395
- Liebeck, Martin W.; Saxl, Jan (1986), "The finite primitive permutation groups of rank three", The Bulletin of the London Mathematical Society 18 (2): 165–172, doi:10.1112/blms/18.2.165, ISSN 0024-6093, MR 818821