- Source: Rank 3 permutation group
In mathematical finite group theory, a 3/info/rank" target="_blank">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 3/info/rank" target="_blank">rank 3 permutation groups.
Classification
The primitive 3/info/rank" target="_blank">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.
Examples
If G is any 4-transitive group acting on a set S, then its action on pairs of elements of S is a 3/info/rank" target="_blank">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 3/info/rank" target="_blank">rank 3 permutation groups.
The projective general linear group acting on lines in a projective space of dimension at least 3 is a 3/info/rank" target="_blank">rank-3 permutation group.
Several 3-transposition groups are 3/info/rank" target="_blank">rank-3 permutation groups (in the action on transpositions).
It is common for the point-stabilizer of a 3/info/rank" target="_blank">rank-3 permutation group acting on one of the orbits to be a 3/info/rank" target="_blank">rank-3 permutation group. This gives several "chains" of 3/info/rank" target="_blank">rank-3 permutation groups, such as the Suzuki chain and the chain ending with the Fischer groups.
Some unusual 3/info/rank" target="_blank">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.
Notes
References
Bannai, Eiichi (1971–72), "Maximal subgroups of low 3/info/rank" target="_blank">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 3/info/rank" target="_blank">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 3/info/rank" target="_blank">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 3/info/rank" target="_blank">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 3/info/rank" target="_blank">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 3/info/rank" target="_blank">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