2008 April

Prove that $S_4$ is solvable but not nilpotent.

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

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

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 $p$.

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 $G$.