Abelian and Solvable Groups

We begin this chapter by looking at commutators and the commutator subgroup of a group. The commutator subgroup of a non-abelian group is a normal subgroup that is not the trivial group. In the second section, on abelian groups, a classification (i.e. a complete list) of finitely generated abelian groups is given. The last section introduces solvable groups, which are important in connection with field extensions in Galois theory. We do not go into field extensions here, but solvable groups are also interesting within group theory.