Exercises

1. Prove that the inverse of a commutator is a commutator.

2. Let $G$ be a group and $G'$ be its commutator subgroup. Prove: $G$ is abelian if and only if $G' = \{e\}$.

3. Write the element $(1,2,3)\in S_3$ as a commutator of elements of $S_3$.

4. We can define the sequence of subgroups, called the derived series, inductively as follows:
   

   $$G^{(n+1)}\subset G^{(n)} \subset ...\subset G''\subset G'\subset G,$$ (8.2)

   
   
   where $G'=G^{(1)}$ is the commutator subgroup of $G$, and $G^{(i+1)}$ is the commutator subgroup of $G^{(i)}$, for $i>0$. Prove that every term in the derived series, (8.2) above, is normal in $G$. (HINT: Theorem 8.1.5.)

5. Find $G'$ if $G = S_3$. Prove that $S_3$ is solvable. (HINT: $A_3$.)

6. Prove that $S_n$ is not solvable for all $n \geq 5$. (What do you think about $S_4$?)