---
tag: Y10S0751KG
kind: concept
name: Primitive permutation group
msc: [20B15]
specializes: [ZDJQRKPWGA]
sources:
  - url: https://en.wikipedia.org/wiki/Primitive_permutation_group
same_as:
  wikidata: Q7243579
  wikipedia: Primitive permutation group
---

## Summary

A permutation group is primitive when it is [[transitive|02GK6HZYBT]] and preserves no nontrivial [[block system|SF738NKWJ0]], that is, no nontrivial splitting of the set into parts.
Primitive groups are the building blocks of all finite permutation groups.
A transitive group of degree $n$ that is not primitive embeds in a [[wreath product|W1X984PKJ9]] $S_k \wr S_m$ with $km = n$ and $1 < k, m < n$.
A transitive group is primitive exactly when its point [[stabilisers|0AYMP41THN]] are maximal subgroups.
The [[O'Nan–Scott theorem|Z4WVANZPK4]] sorts the finite primitive groups into a few types.

## Explanations

### L1 human

Imagine a group that shuffles a set of points.
If the group always keeps some grouping of the points together, for example pairs of opposite corners of a square, then it is not primitive.
A primitive group has no such hidden grouping: the only groupings it respects are the trivial ones, all points in one part or each point alone.
The full group of all permutations of a set is primitive.
The symmetries of a square, acting on its corners, are not primitive, because they keep the two diagonals as pairs.