# Group Theory/Subnormal subgroups and series

{{definition|subnormal subgroup|Let be a group. A subgroup is called **subnormal subgroup** if and only if there exists

**Definition (subnormal series)**:

Let be a group. Then a **subnormal series** is a finite family of subgroups such that

- ,

where is the identity.

**Definition (composition series)**:

Let be a group. A **composition series** of is a subnormal series

of such that for all the quotient group is simple.

**Theorem (Schreier refinement theorem)**:

Let be a group, and let

be a subnormal series of .