Problem :

Let G be a ﬁnite group acting on a ﬁnite set. Prove that if for two elements then , where for any element .

My attempt:

Let A be the cyclic subgroup of G generated by a and B the cyclic subgroup of G generated by b.

Since A = B, we know that and .

Let N be the number of orbit in A. It is the same number as in B.

From the Cauchy-Frobenius formula (or Burnside it seems there is some disputes about who found it first), .

By the definition of I(g), if then . Else, then . Therefore, we are sure that if then since for some .

Is that correct?