# Group Theory/Subnormal subgroups and series

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

Definition (subnormal series):

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

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

where ${\displaystyle e\in G}$ is the identity.

Definition (composition series):

Let ${\displaystyle G}$ be a group. A composition series of ${\displaystyle G}$ is a subnormal series

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

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

Theorem (Schreier refinement theorem):

Let ${\displaystyle G}$ be a group, and let

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

be a subnormal series of ${\displaystyle G}$.