Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ?

share|cite|improve this question
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 $x\mapsto g^{-1}x$)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.