## Group Theory Exam

Reading time: less than a minute (114 words).

## 2003 November

1. Let $H$ be a finite index subgroup of $G$. Show that there exists a finite index subgroup $K$ of $G$ such that $K \subseteq H$ and $K$ is normal in $G$.

2. Let $G$ be a group of order 84. Show that $G$ is not simple.

3. Let $G$ be a group and let $A$ be an abelian normal subgroup of $G$. Show that there is a nontrivial homomorphism from $G/A$ to the automorphism group of $A$.

4. State what it means for a group to be solvable, and show that any group of order 280 is solvable.

5. Prove that a group of order 343 has a nontrivial center.