Abstract Algebra: Theory and Applications

A subnormal (normal) series $\{K_j\}$ is a refinement of a subnormal (normal) series $\{H_i\}$ if $\{H_i\}\subset \{K_j\}\text{.}$ That is, each $H_i$ is one of the $K_j\text{.}$

The best way to study a subnormal or normal series of subgroups, $\{H_i\}$ of $G\text{,}$ is actually to study the factor groups $H_{i+1}/H_i\text{.}$ We say that two subnormal (normal) series $\{H_i\}$ and $\{K_j\}$ of a group $G$ are isomorphic if there is a one-to-one correspondence between the collections of factor groups $\{H_{i+1}/H_i\}$ and $\{K_{j+1}/K_j\}\text{.}$

A subnormal series $\{H_i\}$ of a group $G$ is a composition series if all the factor groups are simple; that is, if none of the factor groups of the series contains a normal subgroup. A normal series $\{H_i\}$ of $G$ is a principal series if all the factor groups are simple.

Although composition series need not be unique as in the case of ${\mathbb{Z}}_{60}\text{,}$ it turns out that any two composition series are related. The factor groups of the two composition series for ${\mathbb{Z}}_{60}$ are ${\mathbb{Z}}_2\text{,}$ ${\mathbb{Z}}_2\text{,}$ ${\mathbb{Z}}_3\text{,}$ and ${\mathbb{Z}}_5\text{;}$ that is, the two composition series are isomorphic. The Jordan-Hölder Theorem says that this is always the case.

A group $G$ is solvable if it has a subnormal series $\{H_i\}$ such that all of the factor groups $H_{i+1}/H_i$ are abelian. Solvable groups will play a fundamental role when we study Galois theory and the solution of polynomial equations.