# Multiply set-transitive group action

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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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## Contents

## Definition

### Symbol-free definition

A group action on a set is termed -**set-transitive** or -**homogeneous** for if the following is true for all :

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

A group action is termed **multiply set-transitive** or **multiply homogeneous** if it is -set-transitive for some .

If a group action is -set-transitive but not -set-transitive, it is said to be sharply -set-transitive.

## Facts

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