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).