Subnormal series

Let $G$ be a group . A subnormal series of $G$ is a finite chain of subgroups $\{e\}=G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$ such that each $G_{i-1}$ is a normal subgroup of $G_i$.

Subnormal series are used to organize inductive arguments on group structure. Important special cases include the derived series and the lower central series . A subnormal series whose factors satisfy additional properties can be a composition series .

Examples:

• The two-step chain $\{e\}\lhd G$ is a subnormal series for every group $G$.
• In $S_4$, the chain $\{e\}\lhd V_4\lhd A_4\lhd S_4$ is a subnormal series (here $V_4$ is the Klein four subgroup).
• In $S_3$, the chain $\{e\}<\langle (12)\rangle<S_3$ is not a subnormal series because $\langle(12)\rangle$ is not normal in $S_3$.