Multiply transitive group action

<h2 id="examples">Examples</h2>
<p>Every group is trivially 1-transitive, by its action on itself by left-multiplication.
</p><p>Let 
  
    
      
        
          S
          
            n
          
        
      
    
    {\displaystyle S_{n}}
  
 be the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> acting on 
  
    
      
        {
        1
        ,
        .
        .
        .
        ,
        n
        }
      
    
    {\displaystyle \{1,...,n\}}
  
, then the action is sharply n-transitive.
</p><p>The group of n-dimensional <a href="/facts/Similarity_(geometry)/Sbd6hEAN">similarities</a> acts 2-transitively on 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
.  In the case 
  
    
      
        n
        =
        1
      
    
    {\displaystyle n=1}
  
 this action is sharply 2-transitive, but for 
  
    
      
        n
        >
        1
      
    
    {\displaystyle n>1}
  
 it is not.
</p><p>The group of n-dimensional <a href="/facts/Homography/wCkNzUjd">projective transforms</a> <i>almost</i> acts sharply (n+2)-transitively on the n-dimensional <a href="/facts/Real_projective_space/svbABDtd">real projective space</a> 
  
    
      
        
          
            R
            P
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {RP} ^{n}}
  
. The <i>almost</i> is because the (n+2) points must be in <a href="/facts/General_linear_position/p0YpbkJv">general linear position</a>. In other words, the n-dimensional projective transforms act transitively on the space of <a href="/facts/Projective_frame/7CMfl4Hj">projective frames</a> of 
  
    
      
        
          
            R
            P
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {RP} ^{n}}
  
.
</p>
<h2 id="classifications-of-2-transitive-groups">Classifications of 2-transitive groups</h2>
<p>Every 2-transitive group is a <a href="/facts/Primitive_group/Qq1nyEwq">primitive group</a>, but not conversely. Every <a href="/facts/Zassenhaus_group/V4WRhmyG">Zassenhaus group</a> is 2-transitive, but not conversely.  The <a href="/facts/Solvable_group/Na7lKIns">solvable</a> 2-transitive groups were classified by <a href="/facts/Bertram_Huppert/eEL3KWHQ">Bertram Huppert</a> and are described in the <a href="/facts/List_of_transitive_finite_linear_groups/90VlqFLb">list of transitive finite linear groups</a>.  The insoluble groups were classified by (Hering 1985) using the <a href="/facts/Classification_of_finite_simple_groups/nMwLJf1K">classification of finite simple groups</a> and are all <a href="/facts/Almost_simple_group/l8lWJWvR">almost simple groups</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Multiply_transitive_group/ExjDPjWa">Multiply transitive group</a></li></ul>

<ul><li>Dixon, John D.; Mortimer, Brian (1996), <a href="https://archive.org/details/permutationgroup0000dixo"><i>Permutation groups</i></a>, Graduate Texts in Mathematics, vol. 163, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-94599-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1409812">1409812</a></li>
<li>Hering, Christoph (1985), "Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II", <i>Journal of Algebra</i>, 93 (1): 151–164, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2F0021-8693%2885%2990179-6">10.1016/0021-8693(85)90179-6</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0021-8693">0021-8693</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0780488">0780488</a></li>
<li><a href="/facts/Bertram_Huppert/eEL3KWHQ">Huppert, Bertram</a> (1957), "Zweifach transitive, auflösbare Permutationsgruppen", <i><a href="/facts/Mathematische_Zeitschrift/8JqT7561">Mathematische Zeitschrift</a></i>, 68: 126–150, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01160336">10.1007/BF01160336</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0094386">0094386</a></li>
<li>Huppert, Bertram; Blackburn, Norman (1982), <i>Finite groups. III.</i>, Grundlehren der Mathematischen Wissenschaften, vol. 243, Berlin-New York: Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-10633-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0650245">0650245</a></li>
<li>Johnson, Norman L.; Jha, Vikram; Biliotti, Mauro (2007), <i>Handbook of finite translation planes</i>, Pure and Applied Mathematics, vol. 289, Boca Raton: Chapman & Hall/CRC, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-58488-605-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2290291">2290291</a></li></ul>

Multiply transitive group action open-in-new

Multiply transitive group action