# Triply transitive group action

This article defines a group action property or a property of group actions: a property that can be evaluated for a group acting on a set.
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## Definition

### Symbol-free definition

A group action on a set is termed triply transitive or 3-transitive if the following two conditions are true:

1. Given any two ordered pairs of distinct elements from the set, there is a group element taking one ordered pair to the other.
2. Given any two ordered triples of pairwise distinct elements from the set, there is a group element taking one ordered triple to the other.

Note that both by definition and by convention, actions on sets of size zero or one are always considered triply transitive. The action on a set of size two is triply transitive if and only if it is transitive.

Note that this is the case $k = 3$ of the general notion of a $k$-transitive, or multiply transitive group action.

### Definition with symbols

Suppose $G$ acts on a set $S$. The action is transitive if the following two conditions are held:

1. For $x \ne y$ and $x' \ne y'$, with $x,x',y,y' \in S$, there exists $g \in G$ such that $g \cdot x = x', g \cdot y = y'$.
2. For $a,b,c$ pairwise distinct in $S$ and $a',b',c'$ pairwise distinct in $S$, there exists $g \in G$ such that $g \cdot a = a', g \cdot b = b', g \cdot c = c'$.

## Examples

• The symmetric group on a set of any size is $k$-transitive for all natural numbers $k$. In particular, it is always triple transitive.
• The alternating group on a set of size five or more is triply transitive.
• Let $F$ be a field. The projective general linear group $PGL(2,F)$ acts naturally on the one-dimensional projective line over $F$. This action is triply transitive.