---
tag: SF738NKWJ0
kind: concept
name: Block system
msc: [20B05, 20B15]
sources:
  - url: https://en.wikipedia.org/wiki/Block_(permutation_group_theory)
---

## Summary

A block system for a group acting on a set is a partition of the set into parts that the group permutes among themselves.
Each part is a block: a set that every group element either maps onto itself or moves off itself entirely.
Block systems measure how far a [[transitive action|02GK6HZYBT]] is from being [[primitive|Y10S0751KG]].
For a transitive action, block systems correspond to subgroups between a point [[stabiliser|0AYMP41THN]] and the whole group.

## Explanations

### L1 human

Picture the four corners of a square and the eight symmetries of the square.
Every symmetry maps a diagonal to a diagonal: the pair of opposite corners stays a pair of opposite corners.
So the two diagonals form a block system: the symmetries may swap the two diagonals, but never break one apart.
The two trivial block systems, one part with everything or every point alone, exist for every action.
A group is primitive when these trivial ones are the only block systems.