Group Theory/Subnormal subgroups and series

{{definition|subnormal subgroup|Let $G$ be a group. A subgroup $H\in G$ is called subnormal subgroup if and only if there exists

Definition (subnormal series):

Let $G$ be a group. Then a subnormal series is a finite family of subgroups $H_{0},H_{1},\ldots ,H_{n}\leq G$ such that

$\{e\}=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G$,

where $e\in G$ is the identity.

Definition (composition series):

Let $G$ be a group. A composition series of $G$ is a subnormal series

$\{e\}=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G$

of $G$ such that for all $k\in \{1,\ldots ,n\}$ the quotient group $H_{k}/H_{k-1}$ is simple.

Theorem (Schreier refinement theorem):

Let $G$ be a group, and let

$\{e\}=H_{0}\triangleleft H_{1}\triangleleft \cdots \triangleleft H_{n}=G$

be a subnormal series of $G$.