Definition:Refinement of Normal Series

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

Let $\left \langle {G_i}\right \rangle_{i \in \left[{0 .. n}\right]}$ be a normal series for $G$:
 * $\left \langle {G_i}\right \rangle_{i \in \left[{0 .. n}\right]} = \left({\left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_{n-1} \triangleleft G_n = G}\right)$

Let $\left \langle {H_j}\right \rangle_{j \in \left[{0 .. m}\right]}$ be another normal series for $G$:
 * $\left \langle {H_j}\right \rangle_{j \in \left[{0 .. m}\right]} = \left({\left\{{e}\right\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_{m-1} \triangleleft H_m = G}\right)$

such that $\left \langle {G_i}\right \rangle_{i \in \left[{0 .. n}\right]} \subseteq \left \langle {H_j}\right \rangle_{j \in \left[{0 .. m}\right]}$

Then $\left \langle {H_j}\right \rangle_{j \in \left[{0 .. m}\right]}$ is a refinement of $\left \langle {G_i}\right \rangle_{i \in \left[{0 .. n}\right]}$.

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.