examples of primitive groups that are not doubly transitive

The group $\mathcal{D}_{2n},n\geq 3$ , the dihedral group of order $2n$ , is the symmetry group of the regular $n$ -gon. (Note that we use the more common notation $\mathcal{D}_{2n}$ for this group rather than $\mathcal{D}_{n}$ ).

$\mathcal{D}_{2n}$ is clearly not doubly transitive for $n\geq 4$ , since it preserves “adjacency” in the vertices. Thus, for example, clearly no element of $\mathcal{D}_{2n}$ can take $(1,2)$ to $(1,3)$ . ( $\mathcal{D}_{2\cdot 3}=\mathcal{D}_{6}$ , the symmetry group of the triangle, is, however, doubly transitive).

We show that for $p$ prime, $\mathcal{D}_{2p}$ 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 $\{0,\ldots,p-1\}$ , and let $r$ be the element of $\mathcal{D}_{2n}$ that takes each vertex into its successor $\pmod{p}$ . Now, suppose a block contains two distinct elements $a, b$ ; assume wlog that $b\neq 0$ . Iteratively apply $\displaystyle r^{b-a}$ to these elements to get

$a$	$b$
$b$	$2b-a$
$2b-a$	$3b-a$
$\ldots$	$\ldots$

Since blocks are either equal or disjoint, we see that the block in question contains $a, b$ , and $nb-a$ for each $n$ . But $a\neq b$ , so $nb-a$ runs through all residues (http://planetmath.org/ResidueSystems) $\pmod{p}$ and thus the block contains each vertex. Thus $D_{2p}$ is primitive.

For nonprime $n$ , $\mathcal{D}_{2n}$ 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