Group Theory Exam

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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.