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Rank 3 permutation group - Wikipedia

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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 T × T ≤ G ≤ T 0 wr ⁡ Z / 2 Z {\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.

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

^ 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)], 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

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