examples of primitive groups that are not doubly transitive

The group π’Ÿ2⁒n,nβ‰₯3, the dihedral groupMathworldPlanetmath of order 2⁒n, is the symmetry group of the regularPlanetmathPlanetmathPlanetmath n-gon. (Note that we use the more common notation π’Ÿ2⁒n for this group rather than π’Ÿn).

π’Ÿ2⁒n is clearly not doubly transitive for nβ‰₯4, since it preserves β€œadjacency” in the vertices. Thus, for example, clearly no element of π’Ÿ2⁒n can take (1,2) to (1,3). (π’Ÿ2β‹…3=π’Ÿ6, the symmetry group of the triangle, is, however, doubly transitive).

We show that for p prime, π’Ÿ2⁒p 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 {0,…,p-1}, and let r be the element of π’Ÿ2⁒n that takes each vertex into its successorMathworldPlanetmathPlanetmathPlanetmath (modp). Now, suppose a block contains two distinct elements a,b; assume wlog that bβ‰ 0. Iteratively apply rb-a to these elements to get

a b
b 2⁒b-a
2⁒b-a 3⁒b-a
… …

Since blocks are either equal or disjoint, we see that the block in question contains a,b, and n⁒b-a for each n. But aβ‰ b, so n⁒b-a runs through all residues (http://planetmath.org/ResidueSystems) (modp) and thus the block contains each vertex. Thus D2⁒p is primitive.

For nonprime n, π’Ÿ2⁒n is not primitive. In this case, if d is a divisor of n, then the set of vertices that are multiples of d form a block.

Title examples of primitive groups that are not doubly transitive
Canonical name ExamplesOfPrimitiveGroupsThatAreNotDoublyTransitive
Date of creation 2013-03-22 17:22:37
Last modified on 2013-03-22 17:22:37
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 7
Author rm50 (10146)
Entry type Example
Classification msc 20B15