## Group Theory Exam

Reading time: less than a minute (150 words).

## 2006 November

1. Find (up to isomorphism) all groups with at most fifteen elements.

2. Let $G$ be a finitely generate group.
a. Prove that every proper subgroup $H < G$ is contained in a maximal proper subgroup.
b. Show that the intersection of the maximal proper subgroups of $G$ is a normal subgroup.

3. a. Carefully state (without proof) the three Sylow theorems.
b. Prove that every group of order $p^2 q$, where $p$ and $q$ are primes with $p< q$ and $p$ does not divide $q-1$, is abelian.

4. Prove that the center of a finite abelian $p$-group is nontrivial.

5. Let $G$ be a group and $H$ a subgroup of $G$, $N$ a normal subgroup of $G$. Show that $H N$ is a subgroup of $G$, that $H\cap N$ is a normal subgroup of $H$, and that $HN/N \cong H/H\cap N$.