Exercises

1. Prove that if $\vert G\vert = p^n$, for some $n \in \mathbb{N}$, and $p$ a prime, then $G$ is a $p$-group.

2. Let $G$ be a finite group. Prove that any $H \leq G$ such that $H$ is a $p$-group (Note from Theorem 10.2.6 $H$ is a prime power group, i.e., $\vert H\vert = p^n$.) is contained in at least one $p$-Sylow subgroup ($p$-Sylow subgroup are maximal p-subgroups in this sense).

   (HINT: Use a double coset decomposition similar to the arguments used in the proofs of the second and third Sylow Theorems, but this time decompose $G$ with respect to H and a $p$-Sylow subgroup.)

3. Let $G$ be a finite group.

   (a) Show that every subgroup $H$ of $G$ which contains the normalizer of a $p$-Sylow subgroup is its own normalizer.

   (HINT: Suppose $N_G(P)\subset H$, where $P$ is a $p$-Sylow subgroup. Now $P$ is a $p$-Sylow subgroup of H (Why?). Let $x \in N(H)$. We need to show $x \in H$ to be finished (Why?). Note first that $xPx^{-1}$ is also in $p$-Sylow subgroup of $H$ (Why?). Now use the second Sylow Theorem applied to the above Sylow subgroups of $H$. From this and the fact that $N_G(P)\subset H$ establish the desired result.)

   (b) Use (a) to show that if $P$ is a $p$-Sylow subgroup then $N(N(P))=N(P)$.

4. Show that if a group has $1 + p$ Sylow subgroups of order $p^a$, then any $2$ of these subgroups have just $p^{a- 1}$ elements in common.

   (HINT: Suppose $P_1$ and $P_2$ are any $2$ $p$-Sylow subgroup's. Use an argument with double cosets decomposing the group $G$ into double cosets with respect to $P_1$ and $N_G(P_2)$ like the argument given for the third Sylow Theorem 10.2.2. Finally, use the second Sylow theorem 10.2.1)

5. Show that if a group $G$ has $1 + p$ Sylow subgroups of order $p^a$, then $G$ contains $p^{a+1}$ elements whose orders are divisors of $p^a$.

   (HINT: Use the result of exercise 4 above. Also you may assume that any two of the $p$-Sylow subgroup's of $G$ intersect in the same subgroup of $G$.)