Exercise 31, § 6

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