## Group Theory Exam

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## 2008 April

1. Prove that $S_4$ is solvable but not nilpotent.

2. List all abelian groups of order 100. Prove that your list is complete.

3. How man elements of order 7 are there in a group of order 168?

4. Let $G' < G$ be the commutator subgroup of a finite group $G$. Let $Z$ be the center of $G$. Let $p$ be a prime number. Suppose that $p$ divides $|Z|$ but does not divide $|Z\cap G'|$. Show that $G$ has a subgroup of index $p$.

5. Assume $G$ is a finite group, $p$ the smallest prime dividing the order of $G$, and $H$ a subgroup of index $p$ in $G$. Prove that $H$ is a normal subgroup of $G$.