This homework set includes problems from chapters 5, 6, and 8.

1. Prove: If G₁ and G₂ are finite cyclic groups of the same order then G₁ and G₂ are isomorphic.

2. Prove: If G is a cyclic group whose order is not a prime, G contains a subgroup H which is neither G nor the identity subgroup. Note that “not a prime” includes the possibility that G is infinite.

3. Suppose NvG and suppose N ≤ H ≤ G. Prove NvH .

4. Let NvG and HvN . Find a counterexample to HvG . Hint A₄, Klein 4-group and a subset of the Klein 4-group.

5. Let G be a cyclic group with a subgroup H. Show that H is a normal subgroup. Show that H is a cyclic subgroup. Show that G/H is cyclic.

6. Suppose G is an abelian, simple group. Prove |G|=p, a prime.

Hint: First consider infinite G and consider any non-identity element. Then consider any finite group of composite order and again consider any non-identity element.

7. A famous and very difficult to prove theorem says: Every simple group which is not of prime order is of even order. Assume this theorem. Consider the chain:

G U G’ U G’’ U G’’’... Note that this chain might be of length 1. Show (use words) that if G is finite , this chain is finite. We call a group “solvable” if the last subgroup in the sequence is the identity. Show that every group of odd order is solvable.