MathOverflow is a question and answer site for professional mathematicians. 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

That is: suppose G is a profinite group acting 1-transitively (but maybe not regularly) on a set X. Is there a reasonable criterion for when there is a g in G and a point a in X such that the g-orbit of a is infinite?

I wonder if it's enough to have a family (g_i, a_i) of pairs in G times X such that the g_i-orbit of a_i has size at least i.

Also, does anybody study these things much? A google search for "profinite group action" yields only a few hits; "profinite permutation group(s)" yields none.

share|cite|improve this question
You might find this article interesting: I don't know if many people are looking at profinite group actions in complete generality, but there is plenty of work on actions on locally finite rooted trees, for instance. – Colin Reid Jun 20 '11 at 8:52

The answer depends on whether the action map GxX -> X is continuous (where I'm assuming X has the discrete topology). If so, then I think transitivity implies X is finite. If not, then you might as well view G as some abstract infinite group. If X is not discrete, e.g., given by a profinite system of sets, then I think you can have more interesting actions.

share|cite|improve this answer
I think you definitely want to consider sets with inverse limit topologies. I mean, where's the fun in finite actions of profinite groups (I mean, aside from all of Galois theory). – Ben Webster Oct 13 '09 at 15:33
Yes, the case I'm interested in is when X is a projective limit of finite discrete spaces and G acts continuously on X. Another way to think about it is you have a 1-transitive infinite permutation group (G,X) which is a projective limit of finite permutation groups (G_i, X_i). – John Goodrick Oct 14 '09 at 0:01
Your proposed criterion smells a lot like the open problem you mentioned in that other thread: whether profinite groups with elements of arbitrarily large order can be torsion. – S. Carnahan Oct 14 '09 at 2:54
Indeed, it does smell a lot like it. If anybody has a proof that the two statements are actually equivalent, I'd love to see it. – John Goodrick Oct 16 '09 at 20:32
If the action is regular, then your criterion is equivalent to the open problem. – S. Carnahan Oct 18 '09 at 0:25

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.