2006 Nov

2006 November

1. Find (up to isomorphism) all groups with at most fifteen elements.

2. Let \(G\) be a finitely generate group.\ a. Prove that every proper subgroup \(H < G\) is contained in a maximal proper subgroup.\ b. Show that the intersection of the maximal proper subgroups of \(G\) is a normal subgroup.

3. a. Carefully state (without proof) the three Sylow theorems.\ b. Prove that every group of order \(p^2 q\), where \(p\) and \(q\) are primes with \(p< q\) and \(p\) does not divide \(q-1\), is abelian.

4. Prove that the center of a finite abelian \(p\)-group is nontrivial.

5. Let \(G\) be a group and \(H\) a subgroup of \(G\), \(N\) a normal subgroup of \(G\). Show that \(H N\) is a subgroup of \(G\), that \(H\cap N\) is a normal subgroup of \(H\), and that \(HN/N \cong H/H\cap N\).