\begin{definition}[Definition:Refinement of Normal Series]
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.
\end{definition}