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 $T\times T\leq G\leq T_{0}\operatorname {wr} Z/2Z$ 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 = 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	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	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+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

^ 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

[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.

[1]