Multiply set-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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VIEW RELATED: group action property implications | group action property non-implications | {{{context space}}} metaproperty satisfactions | group action metaproperty dissatisfactions | group action property satisfactions |group action property dissatisfactions

Definition

Symbol-free definition

A group action on a set is termed $k$ -set-transitive or $k$ -homogeneous for $k\geq 1$ if the following is true for all $1\leq l\leq k$ :

Consider the natural action of the group on the set of subsets of size $l$ . This action is a transitive group action.

A group action is termed multiply set-transitive or multiply homogeneous if it is $k$ -set-transitive for some $k>1$ .

If a group action is $k$ -set-transitive but not $(k+1)$ -set-transitive, it is said to be sharply $k$ -set-transitive.

Facts

The symmetric group on any set is $k$ -set-transitive on it for every $k$ .
The alternating group on any finite set (or more generally, the finitary alternating group on any set) is $k$ -set-transitive on it for every $k$ .

Relation with other properties

Stronger properties