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In [[mathematics]], especially in areas of [[abstract algebra]] and [[finite geometry]], the '''list of transitive finite linear groups''' is an important classification of certain highly symmetric [[group action|actions]] of [[finite group]]s on [[vector space]]s.
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The [[solvable group|solvable]] finite [[2-transitive group]]s were classified by [[Bertram Huppert]].<ref>{{Citation | last1=Huppert | first1=Bertram | author1-link=Bertram Huppert | title=Zweifach transitive, auflösbare Permutationsgruppen | doi=10.1007/BF01160336 | id={{MathSciNet | id = 0094386}} | year=1957 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=68 | pages=126–150}}</ref> The [[classification of finite simple groups]] made possible the complete classification of finite [[doubly transitive]] [[permutation groups]]. This is a result by [[Christoph Hering]].<ref>{{Citation | last1=Hering | first1=Christoph | title=Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II | doi=10.1016/0021-8693(85)90179-6 | id={{MathSciNet | id = 780488}} | year=1985 | journal=Journal of Algebra | issn=0021-8693 | volume=93 | issue=1 | pages=151–164}}</ref>  A finite 2-transitive group has a [[socle (mathematics)|socle]] that is either a vector space over a [[finite field]] or a non-abelian primitive [[simple group]]; groups of the latter kind are almost simple groups and described elsewhere<!-- haha, where? someday at [[almost simple group]]? -->. This article provides a complete <!-- and irredundant? --> list of the finite 2-transitive groups whose socle is [[elementary abelian group|elementary abelian]].
 
Let <math>p</math> be a prime, and <math>G</math> a subgroup of the [[general linear group]] <math>GL(d,p)</math> acting transitively on the nonzero vectors of the ''d''-dimensional vector space <math>(F_p)^d</math> over the finite field <math>F_p</math> with ''p'' elements.
 
== Infinite classes ==
 
There are four infinite classes of finite transitive linear groups.
 
* <math>G \leq \Gamma{}L(1,p^d);</math>
* <math>G \triangleright SL(a,q)\text{ and }p^d=q^a;</math>
* <math>G \triangleright Sp(2a,q)\text{ and }p^d=q^{2a};</math>
* <math>G \triangleright G_2(q)',\  p^d=q^6\text{ and }p=2.</math>
 
Notice that the exceptional [[group of Lie type]] ''G''<sub>2</sub>(''q'') is usually constructed as the automorphism groups of the split [[octonion]]s. Hence, it has a natural [[Representation (mathematics)|representation]] as a subgroup of the 7-dimensional [[orthogonal group]] O(7,&nbsp;''q''). If ''q'' is even, then the underlying [[quadratic form]] polarizes to a degenerate [[symplectic form]]. Factoring out with the radical, one obtains an [[isomorphism]] between O(7,&nbsp;''q'') and the [[symplectic group]] Sp(6,&nbsp;''q''). The subgroup of Sp(6,&nbsp;''q'') which corresponds to ''G''<sub>2</sub>(''q'')′ is transitive.
 
In fact, for ''q''>2, the group ''G''<sub>2</sub>(''q'') = ''G''<sub>2</sub>(''q'')′ is simple. If ''q''=2 then ''G''<sub>2</sub>(2)′ ≅ PSU(3,3) is simple with index 2 in ''G''<sub>2</sub>(2).
 
== Sporadic finite transitive linear groups ==
 
These groups are usually classified by some typical [[normal subgroup]], this normal subgroup is denoted by ''G''<sub>0</sub> and are written in the third column of the table. The notation 2<sup>1+4</sup> stands for the [[extraspecial group]] of order 32.<!-- odd number of quaternion factors or even? -->
 
All but one of the sporadic transitive linear groups <math>G</math> yield a primitive permutation group <math>p^d:G</math> of degree at most 2499. In the computer algebra programs [[GAP computer algebra system|GAP]] and [[Magma computer algebra system|MAGMA]], these groups can be accessed with the command <code>PrimitiveGroup(p^d,k);</code> where the number ''k'' is the ''primitive identification'' of <math>p^d:G</math>. This number is given in the last column of the following table.
 
Seven of these groups are sharply transitive; these groups were found by [[Hans Zassenhaus]] and are also known as the multiplicative groups of the Zassenhaus [[near-field (mathematics)|near-field]]s. These groups are marked by a star in the table.  
 
{| class="wikitable" border="1"
|-
! Condition on <math>p</math>
! Condition on <math>d</math>
! <math>G_0</math>
! Primitive identification of <math>p^d:G</math>
|-
| <math>p=5</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 15*, 18, 19
|-
| <math>p=7</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 25*, 29
|-
| <math>p=11</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 39*, 42
|-
| <math>p=23</math>
| <math>d=2</math>
| <math>SL(2,3)</math>
| 59*
|-
| <math>p=9</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 124, 126, 127, 128
|-
| <math>p=11</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 56*, 57
|-
| <math>p=19</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 86
|-
| <math>p=29</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| 106*, 110
|-
| <math>p=59</math>
| <math>d=2</math>
| <math>SL(2,5)</math>
| no id, * (regular)
|-
| <math>p=3</math>
| <math>d=4</math>
| <math>2^{1+4}</math>
| 71, 90, 99, 129, 130
|-
| <math>p=2</math>
| <math>d=4</math>
| <math>A_6</math>
| 16, 17
|-
| <math>p=2</math>
| <math>d=4</math>
| <math>A_7</math>
| 20
|-
| <math>p=3</math>
| <math>d=6</math>
| <math>SL(2,13)</math>
| 396
|-
|}
 
This list is not explicitly contained in Hering's paper. Many books<ref>{{Citation | last1=Huppert | first1=Bertram | last2=Blackburn | first2=Norman | title=Finite groups. III. | publisher=Springer-Verlag | location=Berlin-New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=3-540-10633-2 | id={{MathSciNet | id = 0650245}} | year=1982 | volume=243}}</ref><ref>{{Citation | last1=Johnson | first1=Norman L. | last2=Jha | first2=Vikram | last3=Biliotti | first3=Mauro | title=Handbook of finite translation planes | publisher=Chapman & Hall/CRC | location=Boca Raton | series=Pure and Applied Mathematics | isbn=978-1-58488-605-1 | id={{MathSciNet | id = 2290291}} | year=2007 | volume=289}}</ref> and papers give a list of these groups, some of them an incomplete one. For example, Cameron's book<ref>{{Citation | last1=Cameron | first1=Peter J. | title=Permutation Groups | publisher=[[Cambridge University Press]] | series=London Mathematical Society Student Texts | isbn=978-0-521-65378-7 | year=1999 | volume=45}}</ref> misses the groups in line 5 of the table.
 
== References ==
{{reflist}}
 
[[Category:Permutation groups]]

Latest revision as of 21:08, 12 August 2014

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