---
tag: AADNYSB6TC
kind: definition
name: Primitive permutation group
msc: [20B15]
defines: Y10S0751KG
uses: [0WCY42EV9V, 02GK6HZYBT, SF738NKWJ0, ZDJQRKPWGA, 0AYMP41THN, BDJ1AX4XBB, 0CB0DX5BVV, 0P7G9NGG2G, 0KVGH0J5B3]
sources:
  - url: https://en.wikipedia.org/wiki/Primitive_permutation_group
---

## Definition

Let a group $G$ [[act|0WCY42EV9V]] on a nonempty set $\Omega$.
The action is *primitive* if:

1. it is [[transitive|02GK6HZYBT]], and
2. the only [[block systems|SF738NKWJ0]] of $\Omega$ for $G$ are the partition into singletons and the partition $\set{\Omega}$ with one part.

A [[permutation group|ZDJQRKPWGA]] $G \le \operatorname{Sym}(\Omega)$ is *primitive* if its natural action on $\Omega$ is primitive.

## Remarks

Condition 2 says that every block $B$ of $G$ has $\abs{B} = 1$ or $B = \Omega$.
If $\abs{\Omega} \ge 2$ and $G$ is transitive, then $G$ is primitive if and only if the [[stabiliser|0AYMP41THN]] $G_\omega$ of a point $\omega$ is a maximal subgroup of $G$.

Example: a transitive group $G$ of prime degree $p$.
For a transitive group, all parts of a block system have the same size, which divides $p$, so each block system is trivial.
For instance, the [[cyclic group|0KVGH0J5B3]] generated by $(1\,2\,3\,4\,5)$ is primitive on $\set{1, \dots, 5}$.
Every $2$-transitive group is primitive (see [[multiply transitive group action|BDJ1AX4XBB]]).

Non-example: the eight symmetries of a square acting on its corners $1, 2, 3, 4$ in cyclic order.
The action is transitive, but the partition $\set{\set{1, 3}, \set{2, 4}}$ into the two diagonals is a nontrivial block system, so clause 2 fails.

If $G$ acts primitively and [[faithfully|0P7G9NGG2G]], then every [[normal subgroup|0CB0DX5BVV]] $N \ne 1$ of $G$ is transitive on $\Omega$, because the orbits of $N$ form a block system.