alternative characterization of multiply transitive permutation groups
For , is -transitive on if and only if for all , is -transitive on .
First assume is -transitive on , and choose . To show is -transitive on , choose . Since is -transitive on , we can choose such that
But obviously , and restricted to is the desired permutation.
To prove the converse, choose . Choose such that
and choose such that
Then is the desired permutation. ∎
Note that this definition of -transitivity affords a straightforward proof of the statement that is -transitive: by inspection, is -transitive; the result follows by induction using the theorem. (The corresponding statement that is -transitive is obvious).
Finally, note that the most common cases of -transitivity are for (transitive), and (doubly transitive).
|Title||alternative characterization of multiply transitive permutation groups|
|Date of creation||2013-03-22 17:21:47|
|Last modified on||2013-03-22 17:21:47|
|Last modified by||rm50 (10146)|