Transitive Group Action -- from Wolfram MathWorld

Transitive Group Action -- from Wolfram MathWorld: "The space , which has a transitive group action, is called a homogeneous space when the group is a Lie group."



(w) Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for every two x, y in X there exists precisely one g in Gsuch that g⋅x = y. In this case, X is called a principal homogeneous space for G or a G-torsor. 

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Group action - Wikipedia

Group action - Wikipedia: "The group action is transitive if and only if it has exactly one orbit, i.e., if there exists x in X with G⋅x = X. This is the case if and only if G⋅x = X for all x in X.

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Orthogonal Group -- from Wolfram MathWorld

Orthogonal Group -- from Wolfram MathWorld: "In fact, the orthogonal group is a smooth -dimensional submanifold"
Because the orthogonal group is a group and a manifold, it is a Lie group. 

... preserve the quadratic form , and so they also preserve circles , which are the group orbits.

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Group Orbit -- from Wolfram MathWorld

Group Orbit -- from Wolfram MathWorld: "For example, consider the action by the circle group on the sphere by rotations along its axis. Then the north pole is an orbit, as is the south pole. The equator is a one-dimensional orbit, as is a general orbit, corresponding to a line of latitude.

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torsors

torsors: " This combination of translations and dilations arises because R is not just a group, but a ring. "



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torsors

torsors: "You can always pretend a torsor is a group. But, it involves an arbitrary choice!"

:

Any group G is a G-torsor, and every other G-torsor is isomorphic to G - but not canonically!



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Group representation - Wikipedia

Group representation - Wikipedia: "formally, a "representation" means a homomorphism from the group to the automorphism group of an object"



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