Exercise 30, § 6

Let $p$ be a prime number, $r$ and $t$ integers $\geqslant 0$, S a cyclic group of order $p^{r+t}$ and T the subgroup of order $p^t$ of S. a) If S operates on a finite set E, now prove that one has

where $E^T$ denotes the set of elements of E invariant under T. b) Let $m, n$ be integers $\geqslant 0$. Now prove that one has (Let M be a set with $m$ elements, and let E be the set of subsets of $S \times M$ having $p^t n$ elements. Define an operation of S on $S \times M$ such that there exists a bijection of $E^T$ onto the set of subsets of $(S/T) \times M$ having $n$ elements; apply a).)