# Definition:Refinement of Normal Series

## Definition

Let $G$ be a group whose identity is $e$.

Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$:

$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$

Let $\sequence {H_j}_{j \mathop \in \closedint 0 m}$ be another normal series for $G$:

$\sequence {H_j}_{j \mathop \in \closedint 0 m} = \tuple {\set e = H_0 \lhd H_1 \lhd \cdots \lhd H_{m - 1} \lhd H_m = G}$

such that $\sequence {G_i}_{i \mathop \in \closedint 0 n} \subseteq \sequence {H_j}_{j \mathop \in \closedint 0 m}$

Then $\sequence {H_j}_{j \mathop \in \closedint 0 m}$ is a refinement of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$.

That is, a refinement of a normal series is a normal series which contains all the (normal) subgroups of the original normal series, and may contain more.

### Proper Refinement

A proper refinement of a normal series is a refinement which is not equal to the original normal series.

That is, it contains extra (normal) subgroups which are not present in the original normal series.