# alternative characterization of multiply transitive permutation groups

This article derives an alternative characterization of $n$-transitive groups.

###### Theorem.

For $n\mathrm{>}\mathrm{1}$, $G$ is $n$-transitive^{} on $X$ if and only if for all $x\mathrm{\in}X$, ${G}_{x}$ is $\mathrm{(}n\mathrm{-}\mathrm{1}\mathrm{)}$-transitive on $X\mathrm{-}\mathrm{\{}x\mathrm{\}}$.

###### Proof.

First assume $G$ is $n$-transitive on $X$, and choose $x\in X$. To show ${G}_{x}$ is $(n-1)$-transitive on $X-\{x\}$, choose ${x}_{1},\mathrm{\dots},{x}_{n-1},{y}_{1},\mathrm{\dots},{y}_{n-1}\in X$. Since $G$ is $n$-transitive on $X$, we can choose $\sigma \in G$ such that

$$\sigma \cdot ({x}_{1},\mathrm{\dots},{x}_{n-1},x)=({y}_{1},\mathrm{\dots},{y}_{n-1},x)$$ |

But obviously $\sigma \in {G}_{x}$, and $\sigma $ restricted to $X-\{x\}$ is the desired permutation^{}.

To prove the converse^{}, choose ${x}_{1},\mathrm{\dots},{x}_{n},{y}_{1},\mathrm{\dots},{y}_{n}\in X$. Choose ${\sigma}_{1}\in {G}_{{x}_{n}}$ such that

$${\sigma}_{1}\cdot ({x}_{1},\mathrm{\dots},{x}_{n-1})=({y}_{1},\mathrm{\dots},{y}_{n-1})$$ |

and choose ${\sigma}_{2}\in {G}_{{y}_{1}}$ such that

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

Then ${\sigma}_{2}{\sigma}_{1}$ is the desired permutation. ∎

Note that this definition of $n$-transitivity affords a straightforward proof of the statement that ${A}_{n}$ is $(n-2)$-transitive: by inspection, ${A}_{3}$ is $1$-transitive; the result follows by induction^{} using the theorem. (The corresponding statement that ${S}_{n}$ is $n$-transitive is obvious).

Finally, note that the most common cases of $n$-transitivity are for $n=1$ (*transitive*), and $n=2$ (*doubly transitive*).

Title | alternative characterization of multiply transitive permutation groups |
---|---|

Canonical name | AlternativeCharacterizationOfMultiplyTransitivePermutationGroups |

Date of creation | 2013-03-22 17:21:47 |

Last modified on | 2013-03-22 17:21:47 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 4 |

Author | rm50 (10146) |

Entry type | Derivation^{} |

Classification | msc 20B20 |

Defines | doubly transitive |