# primitive permutation group

Let $X$ be a set, and $G$ a transitive^{} permutation group^{} on $X$.
Then $G$ is said to be a *primitive permutation group* if it has no nontrivial blocks (http://planetmath.org/BlockSystem).

For example, the symmetric group^{} ${S}_{4}$ is a primitive permutation group on $\{1,2,3,4\}$.

Note that ${D}_{8}$ is not a primitive permutation group on the vertices of a square, because the pairs of opposite points form a nontrivial block.

It can be shown that a transitive permutation group $G$ on a set $X$ is primitive if and only if the stabilizer^{} ${\mathrm{Stab}}_{G}(x)$ is a maximal subgroup of $G$ for all $x\in X$.

Title | primitive permutation group |
---|---|

Canonical name | PrimitivePermutationGroup |

Date of creation | 2013-03-22 14:00:49 |

Last modified on | 2013-03-22 14:00:49 |

Owner | Thomas Heye (1234) |

Last modified by | Thomas Heye (1234) |

Numerical id | 20 |

Author | Thomas Heye (1234) |

Entry type | Definition |

Classification | msc 20B15 |