A friend recently asked me if a finite simple group acts transitively on a set, then is the set finite?
I want to say yes, since if the action is transitive, then the cardinality of the orbit of any element is the cardinality of the whole set, and the cardinality of such an orbit must divide the index of the stabilizer in the group, which is finite since the group is. (I don't think the hypothesis that the group is simple is used at all.)
My main concern is that maybe the orbit stabilizer theorem only applies when the set is already known to be finite. What's the answer here? Thanks.