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 .