1. Let $G$ be a group and let $H \leq G$ be a subgroup. Let $P \in S y l_p(H)$ for some prime $p$. Prove that if $P \unlhd H \unlhd G$, then $P \unlhd G$. (Recall that this statement may be false if $P \notin S y l_p(H)$ !)

Let $G$ be a group of order 36 . Prove that $G$ is not simple.

Let $G$ be a group. Let $|G|=p^2 q^2$, where $p<q$ are distinct primes. Prove that $G$ is not simple. (Hint: Reduce to the case where $p=2, q=3$ ).

Let $G$ be a group. Let $|G|=p q r$, where $p<q<r$ are distinct primes. Prove that $G$ is not simple.

Classify the simple groups of order 2 through 59 . That is, make a list all of the simple groups whose order is between 2 and 59 , inclusive. Then prove that your list is complete.
Hint: The simple groups are just the obvious ones. To prove that your list is complete, you will need to go case by case; that is, show that $Z_2$ is the only simple group of order $2, Z_3$ is the only simple group of order 3 , there are no simple groups of order 4 , etc. While this seems daunting, the vast majority of cases are covered by theorems we have proved in class or on the homework! The only cases that will require any work are 24,40 , and 56 .

Classify the simple groups of order 61 through 100 .
Hint: The only cases that require work are $72,80,84$, and 90 , and the only really tricky one is 90. For 90, consider the Sylow 3-subgroups. If $n_3 \neq 1$, break the problem into 2 cases.
Case 1: For any distinct $P, Q \in \operatorname{Syl}_3(G), P \cap Q={e}$. This case can be ruled out with a simple counting argument.
Case 2: There exist distinct $P, Q \in S y l_3(G)$ such that $P \cap Q \neq{e}$. In this case, let $H=$ $P \cap Q$. Show that $N_G(H)$ has at least 18 elements, so that $\left[G: N_G(H)\right] \leq 5$.

Let $G$ be a group of order $p^2$, where $p$ is a prime.
(a) Prove $G$ is abelian. (Hint: We have proved some useful facts about $p$-groups.)
(b) Prove that $G \cong Z_{p^2}$ or $G \cong Z_p \times Z_p$.

Let $G$ be a group of order $p q$, where $p<q$ are primes and $p \nmid q-1$. Prove that $G \cong Z_{p q}$. (Hint: Recall that $Z_{p q} \cong Z_p \times Z_q$.)

Prove that there is no injective homomorphism from $D_{50}$ to $S_{10}$. (Hint: What does a Sylow 5 -subgroup of $S_{10}$ look like?)

Construct a nonabelian group of order 55. After you have constructed the group, write down a presentation for it! (Hint: The techniques you used in Problem 10 of Homework 8 will be helpful.)