---
tag: 0QP2YYR79N
kind: definition
name: Transitive action
aliases: [transitive-group-action]
msc: [20B05]
defines: 02GK6HZYBT
uses: [0YSE802YCQ, 00F2JFA41J]
sources:
  - url: https://en.wikipedia.org/wiki/Group_action
    locator: Transitivity properties
---

## Definition

Let a [[group|G5S1M9]] $G$ [[act|0WCY42EV9V]] on a nonempty set $X$.
The action is transitive when for all $x, y \in X$ there is a $g \in G$ with $g \cdot x = y$.

## Remarks

The action is transitive if and only if it has exactly one [[orbit|0YQPZB5Q1G]].
The action of $S_n$ on $\{1, \dots, n\}$ is transitive.
The action of $G$ on the left cosets of a subgroup $H$ is transitive. Each transitive action of $G$ is isomorphic to such an action.

## Explanations

### L1 human

A transitive action can move any point to any other point.
So the set does not split into separate pieces under the action.
The rotations of a square act transitively on its four corners.