View source for Rank 3 permutation group

In mathematical [[finite group theory]], a '''rank 3 permutation group''' [[Group action (mathematics)|acts]] transitively on a set such that the [[stabilizer (group theory)|stabilizer]] of a point has 3 [[orbit (group theory)|orbit]]s. The study of these groups was started by {{harvs|txt|authorlink=Donald Higman|last=Higman|year1=1964|year2=1971}}. Several of the [[sporadic simple group]]s were discovered as rank 3 permutation groups.
==Classification==
The primitive rank 3 permutation groups are all in one of the following classes:
*{{harvtxt|Cameron|1981}} classified the ones such that <math>T\times T\le G\le T_0 \operatorname{wr} Z/2Z</math> where the [[Socle (mathematics)|socle]] ''T'' of ''T''<sub>0</sub> is simple, and ''T''<sub>0</sub> is a 2-transitive group of degree {{radic|''n''}}.
*{{harvtxt|Liebeck|1987}} classified the ones with a regular elementary abelian normal subgroup
*{{harvtxt|Bannai|1971–72}} classified the ones whose socle is a simple alternating group
*{{harvtxt|Kantor|Liebler|1982}} classified the ones whose socle is a simple classical group
*{{harvtxt|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.<ref>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.</ref> In particular most of the alternating groups, symmetric groups, and [[Mathieu group]]s 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 group]]s 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 {{harv|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.
{| class="wikitable sortable"
|-
! Group !! Point stabilizer !! size !! Comments
|-
| A<sub>6</sub> = L<sub>2</sub>(9) = Sp<sub>4</sub>(2)' = M<sub>10</sub>' || S<sub>4</sub> || 15 = 1+6+8 || Pairs of points, or sets of 3 blocks of 2, in the 6-point permutation representation; two classes
|-
| A<sub>9</sub> || L<sub>2</sub>(8):3 || 120 = 1+56+63 || Projective line P<sub>1</sub>(8); two classes
|-
| A<sub>10</sub> || (A<sub>5</sub>×A<sub>5</sub>):4 || 126 = 1+25+100 || Sets of 2 blocks of 5 in the natural 10-point permutation representation
|-
| L<sub>2</sub>(8) || 7:2 = Dih(7) || 36 = 1+14+21 || Pairs of points in P<sub>1</sub>(8)
|-
| L<sub>3</sub>(4) || A<sub>6</sub> || 56 = 1+10+45 || Hyperovals in P<sub>2</sub>(4); three classes
|-
| L<sub>4</sub>(3) || PSp<sub>4</sub>(3):2 || 117 = 1+36+80 || Symplectic polarities of P<sub>3</sub>(3); two classes
|-
| G<sub>2</sub>(2)' = U<sub>3</sub>(3) || PSL<sub>3</sub>(2) || 36 = 1+14+21 || [[Suzuki chain]]
|-
| U<sub>3</sub>(5) || A<sub>7</sub> || 50 = 1+7+42 || The action on the vertices of the [[Hoffman-Singleton graph]]; three classes
|-
| U<sub>4</sub>(3) || L<sub>3</sub>(4) || 162 = 1+56+105 || Two classes
|-
| Sp<sub>6</sub>(2) || G<sub>2</sub>(2) = U<sub>3</sub>(3):2 || 120 = 1+56+63 || The Chevalley group of type G<sub>2</sub> acting on the octonion algebra over GF(2)
|-
| Ω<sub>7</sub>(3) || G<sub>2</sub>(3) || 1080 = 1+351+728 || The Chevalley group of type G<sub>2</sub> acting on the imaginary octonions of the octonion algebra over GF(3); two classes
|-
| U<sub>6</sub>(2) || U<sub>4</sub>(3):2<sub>2</sub> || 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
|-
| [[Mathieu group|M<sub>11</sub>]] || M<sub>9</sub>:2 = 3<sup>2</sup>:SD<sub>16</sub> || 55 = 1+18+36 || Pairs of points in the 11-point permutation representation
|-
| [[Mathieu group|M<sub>12</sub>]] || M<sub>10</sub>:2 = A<sub>6</sub>.2<sup>2</sup> = 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
|-
| [[Mathieu group|M<sub>22</sub>]] || 2<sup>4</sup>:A<sub>6</sub> || 77 = 1+16+60 || Blocks of S(3,6,22)
|-
| [[Janko group J2|J<sub>2</sub>]] || U<sub>3</sub>(3) || 100 = 1+36+63 || [[Suzuki chain]]; the action on the vertices of the [[Hall-Janko graph]]
|-
| [[Higman–Sims group|Higman-Sims group HS]] || [[Mathieu group|M<sub>22</sub>]] || 100 = 1+22+77 || The action on the vertices of the [[Higman-Sims graph]]
|-
| [[Mathieu group|M<sub>22</sub>]] || A<sub>7</sub> || 176 = 1+70+105 || Two classes
|-
| [[Mathieu group|M<sub>23</sub>]] || M<sub>21</sub>:2 = L<sub>3</sub>(4):2<sub>2</sub> = PΣL(3,4) || 253 = 1+42+210 || Pairs of points in the 23-point permutation representation
|-
| [[Mathieu group|M<sub>23</sub>]] || 2<sup>4</sup>:A<sub>7</sub> || 253 = 1+112+140 || Blocks of S(4,7,23)
|-
| [[McLaughlin group (mathematics)|McLaughlin group McL]] || U<sub>4</sub>(3) || 275 = 1+112+162 || The action on the vertices of the [[McLaughlin graph]]
|-
| [[Mathieu group|M<sub>24</sub>]] || M<sub>22</sub>:2 || 276 = 1+44+231 || Pairs of points in the 24-point permutation representation
|-
| G<sub>2</sub>(3) || U<sub>3</sub>(3):2 || 351 = 1+126+244 || Two classes
|-
| G<sub>2</sub>(4) || [[Janko group J2|J<sub>2</sub>]] || 416 = 1+100+315 || [[Suzuki chain]]
|-
| [[Mathieu group|M<sub>24</sub>]] || M<sub>12</sub>:2 || 1288 = 1+495+792 || Pairs of complementary dodecads in the 24-point permutation representation
|-
| [[sporadic Suzuki group|Suzuki group Suz]] || G<sub>2</sub>(4) || 1782 = 1+416+1365 || [[Suzuki chain]]
|-
||G<sub>2</sub>(4) || U<sub>3</sub>(4):2 || 2016 = 1+975+1040 ||
|-
|[[Conway group Co2|Co<sub>2</sub>]] || PSU<sub>6</sub>(2):2 || 2300 = 1+891+1408 ||
|-
|[[Rudvalis group|Rudvalis group Ru]] || [[Tits group|<sup>2</sup>F<sub>4</sub>(2)]] || 4060 = 1+1755+2304 ||
|-
|[[Fischer group Fi22|Fi<sub>22</sub>]] || 2.PSU<sub>6</sub>(2) || 3510 = 1+693+2816 || [[3-transposition group|3-transpositions]]
|-
|[[Fischer group Fi22|Fi<sub>22</sub>]] || Ω<sub>7</sub>(3) || 14080 = 1+3159+10920 || Two classes
|-
|[[Fischer group Fi23|Fi<sub>23</sub>]] || 2.[[Fischer group|Fi<sub>22</sub>]] || 31671 = 1+3510+28160 || [[3-transposition group|3-transpositions]]
|-
| G<sub>2</sub>(8).3 || SU<sub>3</sub>(8).6 || 130816 = 1+32319+98496 ||
|-
|[[Fischer group Fi23|Fi<sub>23</sub>]] || PΩ<sub>8</sub><sup>+</sup>(3).S<sub>3</sub> || 137632 = 1+28431+109200 ||
|-
|[[Fischer group Fi24|Fi<sub>24</sub>]]' || [[Fischer group|Fi<sub>23</sub>]] || 306936 = 1+31671+275264 || [[3-transposition group|3-transpositions]]
|-
|}
==Notes==
{{reflist}}
==References==
*{{Citation | last1=Bannai | first1=Eiichi | title=Maximal subgroups of low rank of finite symmetric and alternating groups | mr=0357559 | year=1971–72 | journal=Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics | issn=0040-8980 | volume=18 | pages=475–486}}
*{{Citation | last1=Brouwer | first1=A. E. | last2=Cohen | first2=A. M. | last3=Neumaier | first3=Arnold | title=Distance-regular graphs | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] | isbn=978-3-540-50619-5 | mr=1002568 | year=1989 | volume=18}}
*{{Citation | last1=Cameron | first1=Peter J. | title=Finite permutation groups and finite simple groups | doi=10.1112/blms/13.1.1 | mr=599634 | year=1981 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=13 | issue=1 | pages=1–22| citeseerx=10.1.1.122.1628 }}
*{{Citation | last1=Higman | first1=Donald G. | author1-link=Donald G. Higman | title=Finite permutation groups of rank 3 | doi=10.1007/BF01111335 | mr=0186724 | year=1964 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=86 | issue=2 | pages=145–156| url=https://deepblue.lib.umich.edu/bitstream/2027.42/46298/1/209_2005_Article_BF01111335.pdf | hdl=2027.42/46298 | s2cid=51836896 | hdl-access=free }}
*{{Citation | last1=Higman | first1=Donald G. | author1-link=Donald G. Higman | title=Actes du Congrès International des Mathématiciens (Nice, 1970) | chapter-url=http://mathunion.org/ICM/ICM1970.1/ | publisher=Gauthier-Villars | mr=0427435 | year=1971 | volume=1 | chapter=A survey of some questions and results about rank 3 permutation groups | pages=361–365}}
*{{Citation |authorlink1=William Kantor| last1=Kantor | first1=William M. | last2=Liebler | first2=Robert A. | title=The rank 3 permutation representations of the finite classical groups | doi=10.2307/1998750 | mr=648077 | year=1982 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=271 | issue=1 | pages=1–71| jstor=1998750 | url=http://darkwing.uoregon.edu/~kantor/PAPERS/KantorLiebler.pdf }}
*{{Citation |authorlink1=Martin Liebeck| last1=Liebeck | first1=Martin W. | title=The affine permutation groups of rank three | doi=10.1112/plms/s3-54.3.477 | mr=879395 | year=1987 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=54 | issue=3 | pages=477–516| citeseerx=10.1.1.135.7735 }}
*{{Citation |authorlink1=Martin Liebeck| last1=Liebeck | first1=Martin W. | last2=Saxl | first2=Jan|author2-link=Jan Saxl | title=The finite primitive permutation groups of rank three | doi=10.1112/blms/18.2.165 | mr=818821 | year=1986 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=18 | issue=2 | pages=165–172}}
[[Category:Finite groups]]