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.

Classification

The primitive rank 3 permutation groups are all in one of the following classes:

• Cameron (1981) classified the ones such that ${\displaystyle T\times T\leq G\leq T_{0}\operatorname {wr} Z/2Z}$ 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.

Examples

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.[1] 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.

For each row in the table below, in the grid in the column marked "size", the number to the left of the equal sign is the degree of the permutation group for the permutation group mentioned in the row. In the grid, the sum to the right of the equal sign shows the lengths of the three orbits of the stabilizer of a point of the permutation group. For example, the expression 15 = 1+6+8 in the first row of the table under the heading means that the permutation group for the first row has degree 15, and the lengths of three orbits of the stabilizer of a point of the permutation group are 1, 6 and 8 respectively.

Group	Point stabilizer	size	Comments
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 = 3^2:SD16	55 = 1+18+36	Pairs of points in the 11-point permutation representation
M12	M10:2 = A6.2^2 = 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	2^4:A6	77 = 1+16+60	Blocks of S(3,6,22)
J2	U3(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	2^4: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+891+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

Notes

1. 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.

References

• 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)], vol. 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, CiteSeerX 10.1.1.122.1628, doi:10.1112/blms/13.1.1, ISSN 0024-6093, MR 0599634
• Higman, Donald G. (1964), "Finite permutation groups of rank 3" (PDF), Mathematische Zeitschrift, 86 (2): 145–156, doi:10.1007/BF01111335, hdl:2027.42/46298, ISSN 0025-5874, MR 0186724, S2CID 51836896
• Higman, Donald G. (1971), "A survey of some questions and results about rank 3 permutation groups", Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 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" (PDF), Transactions of the American Mathematical Society, 271 (1): 1–71, doi:10.2307/1998750, ISSN 0002-9947, JSTOR 1998750, MR 0648077
• Liebeck, Martin W. (1987), "The affine permutation groups of rank three", Proceedings of the London Mathematical Society, Third Series, 54 (3): 477–516, CiteSeerX 10.1.1.135.7735, doi:10.1112/plms/s3-54.3.477, ISSN 0024-6115, MR 0879395
• 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 0818821