Exercise 31, § 6

Let G be a finite group, and S a $p$-Sylow subgroup of G. Let $r, t$ be integers $\geqslant 0$ such that $\operatorname{Card}(S) = p^{r+t}$. Let E be the set of subgroups of G of order $p^t$. a) Suppose that S is cyclic. Now prove that $\operatorname{Card}(E) \equiv 1 \pmod{p^{r+1}}$. (Have S act on E by conjugation and use exerc. 30.) Deduce that, if the subgroup of order $p^t$ of S is not normal in G, there are at least $1 + p^{r+1}$ $p$-Sylow subgroups in G. b) Now prove that $\operatorname{Card}(E) \equiv 1 \pmod{p}$ even if S is not cyclic. (Have G act by translation on the set F of subsets of G having $p^t$ elements. Show that the elements of E give distinct orbits having $(G:S)p^r$ elements, and that all the other orbits have a number of elements divisible by $p^{r+1}$. Apply exerc. 30.)