examples of primitive groups that are not doubly transitive
is clearly not doubly transitive for , since it preserves “adjacency” in the vertices. Thus, for example, clearly no element of can take to . (, the symmetry group of the triangle, is, however, doubly transitive).
We show that for prime, is primitive. To prove this, we need only verify that any block containing two distinct elements is the entire set of vertices. Number the vertices consecutively , and let be the element of that takes each vertex into its successor . Now, suppose a block contains two distinct elements ; assume wlog that . Iteratively apply to these elements to get
Since blocks are either equal or disjoint, we see that the block in question contains , and for each . But , so runs through all residues (http://planetmath.org/ResidueSystems) and thus the block contains each vertex. Thus is primitive.
For nonprime , is not primitive. In this case, if is a divisor of , then the set of vertices that are multiples of form a block.
|Title||examples of primitive groups that are not doubly transitive|
|Date of creation||2013-03-22 17:22:37|
|Last modified on||2013-03-22 17:22:37|
|Last modified by||rm50 (10146)|