# examples of primitive groups that are not doubly transitive

The group ${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x81\u2019n},n\beta \x89\u20af3$, the dihedral group^{} of order $2\beta \x81\u2019n$, is the symmetry group of the regular^{} $n$-gon. (Note that we use the more common notation ${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x81\u2019n}$ for this group rather than ${\mathrm{\pi \x9d\x92\x9f}}_{n}$).

${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x81\u2019n}$ is clearly not doubly transitive for $n\beta \x89\u20af4$, since it preserves βadjacencyβ in the vertices. Thus, for example, clearly no element of ${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x81\u2019n}$ can take $(1,2)$ to $(1,3)$. (${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x8b\x853}={\mathrm{\pi \x9d\x92\x9f}}_{6}$, the symmetry group of the triangle, is, however, doubly transitive).

We show that for $p$ prime, ${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x81\u2019p}$ 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 $\{0,\mathrm{\beta \x80\xa6},p-1\}$, and let $r$ be the element of ${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x81\u2019n}$ that takes each vertex into its successor^{} $(modp)$. Now, suppose a block contains two distinct elements $a,b$; assume wlog that $b\beta \x890$. Iteratively apply ${r}^{b-a}$ to these elements to get

$a$ | $b$ |

$b$ | $2\beta \x81\u2019b-a$ |

$2\beta \x81\u2019b-a$ | $3\beta \x81\u2019b-a$ |

$\mathrm{\beta \x80\xa6}$ | $\mathrm{\beta \x80\xa6}$ |

Since blocks are either equal or disjoint, we see that the block in question contains $a,b$, and $n\beta \x81\u2019b-a$ for each $n$. But $a\beta \x89b$, so $n\beta \x81\u2019b-a$ runs through all residues (http://planetmath.org/ResidueSystems) $(modp)$ and thus the block contains each vertex. Thus ${D}_{2\beta \x81\u2019p}$ is primitive.

For nonprime $n$, ${\mathrm{\pi \x9d\x92\x9f}}_{2\beta \x81\u2019n}$ 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 |
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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 |