# doubly transitive groups are primitive

###### Theorem.

Every doubly transitive group is primitive (http://planetmath.org/PrimativeTransitivePermutationGroupOnASet).

###### Proof.

Let $G$ acting on $X$ be doubly transitive. To show the action is , we must show that all blocks are trivial blocks; to do this, it suffices to show that any block containing more than one element is all of $X$. So choose a block $Y$ with two distinct elements ${y}_{1},{y}_{2}$. Given an arbitrary $x\in X$, since $G$ is doubly transitive, we can choose $\sigma \in G$ such that

$$\sigma \cdot ({y}_{1},{y}_{2})=({y}_{1},x)$$ |

But then $\sigma \cdot Y\cap Y\ne \mathrm{\varnothing}$, since ${y}_{1}$ is in both. Thus $\sigma \cdot Y=Y$, so $x\in Y$ as well. So $Y=X$ and we are done. ∎

Title | doubly transitive groups are primitive |
---|---|

Canonical name | DoublyTransitiveGroupsArePrimitive |

Date of creation | 2013-03-22 17:21:50 |

Last modified on | 2013-03-22 17:21:50 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 7 |

Author | rm50 (10146) |

Entry type | Theorem |

Classification | msc 20B15 |