TITLE: When does a transitive action of a profinite group have an infinite orbit?
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.