Multiply 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 $k$-transitive for $k\ge 1$ if the following is true for all $1\le l\le k$:

Consider the set of those ordered $l$-tuples over the set being acted upon that have distinct entries. The group naturally acts on this set of ordered tuples by the action on each coordinate. This action must be a transitive group action.

A group action is termed multiply transitive if it is $k$-transitive for some $k>1$. If a group action is $k$-transitive but not $(k+1)$-transitive, then the group action is termed sharply $k$-transitive.

Facts

• The symmetric group on any set is $k$-transitive for every $k$.
• The alternating group on a set of size $n$ is $(n-2)$-transitive.

Relation with other properties

Stronger properties

• Highly transitive group action
• Triply transitive group action
• Doubly transitive group action

Weaker properties

• Multiply set-transitive group action