---
tag: 5CYEQY4SWR
kind: concept
name: Torsor
msc: [14L15, 14L30, 20G10]
sources:
  - url: https://en.wikipedia.org/wiki/Principal_homogeneous_space
same_as:
  wikidata: Q3755543
  wikipedia: Principal homogeneous space
---

## Summary

A torsor, or principal homogeneous space, for a group $G$ is a space with a [[group action|0WCY42EV9V]] of $G$ that is free and transitive.
It looks exactly like $G$ except that no point is singled out as the identity.
Let $G$ be an [[algebraic group|GH81SYAPPN]] over a field $k$.
A $G$-torsor may have no point over $k$ even though it becomes isomorphic to $G$ over a larger field.
Such torsors are classified by the [[Galois cohomology|9MPCZ3EAJM]] set $H^1(k, G)$, at least when $G$ is smooth.

## Explanations

### L1 human

The points of a line form a torsor for the group of translations.
Two points differ by exactly one translation, but there is no natural origin, so one cannot add points, only move them.
Similarly, the musical notes form a torsor under shifts by intervals, and calendar dates form a torsor under shifts by days.
A torsor is a group that has forgotten which element is the identity.