Doubly 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 said to be doubly transitive or 2-transitive if the induced action on the set of ordered tuples of distinct elements, is transitive. In other words, given any two ordered tuples each having a pair of distinct elements, there is a group element taking one ordered tuple to the other.

Another way of saying this is that the stabilizers of any two points must intersect in a subgroup whose index in each is 1 less than the index of the subgroups.

Definition with symbols

A group action of a group $G$ on a set $S$ is said to be doubly transitive if given any $(x,y)$ and $(x',y')$ with $x\neq y,x'\neq y'$, all elements of $S$, there exists $g\in G$ such that $g.x=x'$ and $g.y=y'$.

Equivalently, it is doubly transitive if it is a transitive group action and if $H$ and $K$ are the stabilizers of two distinct points, and their index is $n$ (which is also the size of $S$, by transitivity) then $H\cap K$ has index $n(n-1)$.

Relation with other properties

Stronger properties

• Triply transitive group action

Weaker properties

• Doubly set-transitive group action
• Primitive group action
• Generously transitive group action
• Transitive group action

Related group properties

• Doubly transitive group is a group possessing a doubly transitive action