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Consider the theorem:

Orbits of a normal subgroup are equal in size when the full group acts transitively

This is stated and proved here. I wonder: Does this theorem also hold infinite sets on which the group acts ? If not, can someone give a counterexample ?

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up vote 3 down vote accepted

More generally, if $O_1, O_2$ are two orbits of $H$, then there exists $g\in G$ such that $x\mapsto gx$ is a bijection $O_1\to O_2$. (The existence follows from $G$ acting transitively, bijectivity by specifying the inverse $x\mapsto g^{-1}x$)

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