# 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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## Definition

### Symbol-free definition

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

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 $k$.