2003 November

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 $G$.

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

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 $A$.

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

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