Jump to content
 







Main menu
   


Navigation  



Main page
Contents
Current events
Random article
About Wikipedia
Contact us
Donate
 




Contribute  



Help
Learn to edit
Community portal
Recent changes
Upload file
 








Search  

































Create account

Log in
 









Create account
 Log in
 




Pages for logged out editors learn more  



Contributions
Talk
 



















Contents

   



(Top)
 


1 Examples  





2 Classifications of 2-transitive groups  





3 See also  





4 References  














Multiply transitive group action






Svenska
 

Edit links
 









Article
Talk
 

















Read
Edit
View history
 








Tools
   


Actions  



Read
Edit
View history
 




General  



What links here
Related changes
Upload file
Special pages
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Wikidata item
 




Print/export  



Download as PDF
Printable version
 
















Appearance
   

 






From Wikipedia, the free encyclopedia
 

(Redirected from Doubly transitive permutation group)

A group acts 2-transitively on a set if it acts transitively on the set of distinct ordered pairs . That is, assuming (without a real loss of generality) that acts on the left of , for each pair of pairs with and , there exists a such that .

The group action is sharply2-transitive if such is unique.

A2-transitive group is a group such that there exists a group action that's 2-transitive and faithful. Similarly we can define sharply2-transitive group.

Equivalently, and , since the induced action on the distinct set of pairs is .

The definition works in general with k replacing 2. Such multiply transitive permutation groups can be defined for any natural number k. Specifically, a permutation group G acting on n points is k-transitive if, given two sets of points a1, ... ak and b1, ... bk with the property that all the ai are distinct and all the bi are distinct, there is a group element ginG which maps aitobi for each i between 1 and k. The Mathieu groups are important examples.

Examples[edit]

Every group is trivially 1-transitive, by its action on itself by left-multiplication.

Let be the symmetric group acting on , then the action is sharply n-transitive.

The group of n-dimensional homothety-translations acts 2-transitively on .

The group of n-dimensional projective transforms almost acts sharply (n+2)-transitively on the n-dimensional real projective space . The almost is because the (n+2) points must be in general linear position. In other words, the n-dimensional projective transforms act transitively on the space of projective framesof.

Classifications of 2-transitive groups[edit]

Every 2-transitive group is a primitive group, but not conversely. Every Zassenhaus group is 2-transitive, but not conversely. The solvable 2-transitive groups were classified by Bertram Huppert and are described in the list of transitive finite linear groups. The insoluble groups were classified by (Hering 1985) using the classification of finite simple groups and are all almost simple groups.

See also[edit]

References[edit]


Retrieved from "https://en.wikipedia.org/w/index.php?title=Multiply_transitive_group_action&oldid=1227185466"

Category: 
Permutation groups
Hidden categories: 
Articles with short description
Short description is different from Wikidata
 



This page was last edited on 4 June 2024, at 07:16 (UTC).

Text is available under the Creative Commons Attribution-ShareAlike License 4.0; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.



Privacy policy

About Wikipedia

Disclaimers

Contact Wikipedia

Code of Conduct

Developers

Statistics

Cookie statement

Mobile view



Wikimedia Foundation
Powered by MediaWiki