The ultimate goal of group theory is to classify all groups up to isomorphism; that is, given a particular group, we should be able to match it up with a known group via an isomorphism. For example, we have already proved that any finite cyclic group of order $n$ is isomorphic to ${\mathbb Z}_n\text{;}$ hence, we “know” all finite cyclic groups. It is probably not reasonable to expect that we will ever know all groups; however, we can often classify certain types of groups or distinguish between groups in special cases.
where each subgroup $H_i$ is normal in $H_{i+1}$ and each of the factor groups $H_{i+1}/H_i$ is abelian, then $G$ is a solvable group. In addition to allowing us to distinguish between certain classes of groups, solvable groups turn out to be central to the study of solutions to polynomial equations.