---
tag: BDJ1AX4XBB
kind: concept
name: Multiply transitive group action
msc: [20B20, 20B22]
specializes: [02GK6HZYBT]
sources:
  - url: https://en.wikipedia.org/wiki/Multiply_transitive_group_action
same_as:
  wikidata: Q4596974
  wikipedia: Multiply transitive group action
---

## Summary

A group action is $k$-transitive when one group element can move any $k$ distinct points onto any other $k$ distinct points, in order.
For $k = 1$ this is a [[transitive action|02GK6HZYBT]], and larger $k$ means a stronger kind of symmetry.
The symmetric group $S_n$ is $n$-transitive and the alternating group $A_n$ is $(n-2)$-transitive on $n$ points, and a $2$-transitive group is always [[primitive|Y10S0751KG]].
Apart from symmetric and alternating groups, finite multiply transitive groups are rare: the [[Mathieu groups|T1KR4KPX8P]] are famous examples.

## Explanations

### L1 human

A group acting on a set is $2$-transitive if, for any two ordered pairs of different points, some element sends the first pair to the second.
Think of a set of seats and a group of allowed ways to reseat people.
If the group is $2$-transitive, any two people can be moved into any two chosen seats.
All permutations of $n$ points form an $n$-transitive group.
The rotations and reflections of a square act transitively on its four corners, but not $2$-transitively: two neighbouring corners cannot be moved onto two opposite corners.