Composition Series and Principal Series

Definition

A composition series of a group $G$ is a subnormal series of finite length

$$1=H_0⊲H_1⊲\cdots ⊲H_n=G,$$

with strict inclusions, such that each $H_i$ is a maximal proper normal subgroup of $H_{i+1}$, i.e., each factor group $H_{i+1}/H_i$ is simple. The factor groups are called composition factors.

The composition series of a finite group is unique in a certain sense, as the following theorem shows.

Jordan-Holder theorem

Any two composition series of a given group are equivalent. That is, they have the same composition length and the same composition factors, up to permutation and isomorphism.

Remark. This theorem is also true for Composition Series of Modules.

Definition

A principal series is a normal series $\{e\}=H_0⊲\cdots ⊲H_m=G$, where $H_i⊲G$ and $H_{i+1}/H_i$ is minimal normal subgroup of $G/H_i$. Remark that $H_{i+1}/H_i$ is called principal factors.