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 $G$:
 * $\left \langle {G_i}\right \rangle_{i \in \left[{0 \,.\,.\, n}\right]} = \left({\left\{{e}\right\} = G_0 \lhd G_1 \lhd \cdots \lhd G_{n-1} \lhd G_n = G}\right)$

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

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.