# alternative characterization of multiply transitive permutation groups

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

###### Theorem.

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

###### 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 $x_{1},\ldots,x_{n-1},y_{1},\ldots,y_{n-1}\in X$. Since $G$ is $n$-transitive on $X$, we can choose $\sigma\in G$ such that

 $\sigma\cdot(x_{1},\ldots,x_{n-1},x)=(y_{1},\ldots,y_{n-1},x)$

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

To prove the converse  , choose $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}\in X$. Choose $\sigma_{1}\in G_{x_{n}}$ such that

 $\sigma_{1}\cdot(x_{1},\ldots,x_{n-1})=(y_{1},\ldots,y_{n-1})$

and choose $\sigma_{2}\in G_{y_{1}}$ such that

 $\sigma_{2}\cdot(y_{2},\ldots,y_{n-1},x_{n})=(y_{2},\ldots,y_{n-1},y_{n})$

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 AlternativeCharacterizationOfMultiplyTransitivePermutationGroups 2013-03-22 17:21:47 2013-03-22 17:21:47 rm50 (10146) rm50 (10146) 4 rm50 (10146) Derivation  msc 20B20 doubly transitive