Principal homogeneous space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Principal_homogeneous_space


00:01:13 1 Definition
00:03:17 2 Examples
00:04:33 3 Applications
00:08:41 4 Other usage
00:11:14 5 See also
00:14:46 6 Notes
00:17:05 7 Further reading
00:17:21 8 External links
00:17:30 Further reading
00:18:42 External links



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SUMMARY
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For the term "torsor" in algebraic geometry, see torsor (algebraic geometry).In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively, meaning that for any x, y in X there exists a unique g in G such that x·g = y where · denotes the (right) action of G on X.
An analogous definition holds in other categories where, for example,

G is a topological group, X is a topological space and the action is continuous,
G is a Lie group, X is a smooth manifold and the action is smooth,
G is an algebraic group, X is an algebraic variety and the action is regular.