It is well known how to compute the number of orbits of a group action. A related problem, apparently not in the literature, is to determine the number of elements in an orbit. The theory that addresses this question leads to orbital extensive categories and to combinatorial aspects of such categories.
Keywords: extensive category, Burnside-Frobenius lemma, conjugacy class of subgroups
2010 MSC: 05A99, 18B99, 20B05
Theory and Applications of Categories, Vol. 30, 2015, No. 17, pp 599-619.