Chief series

Let $G$ be a group . A chief series of $G$ is a finite chain of subgroups $\{e\}=N_0 \lhd N_1 \lhd \cdots \lhd N_r = G$ such that:

1. each $N_i$ is a normal subgroup of $G$, and
2. for each $i$, there is no normal subgroup $K\lhd G$ with $N_{i-1}<K<N_i$.

Equivalently, each factor quotient group $N_i/N_{i-1}$ is a minimal normal subgroup of $G/N_{i-1}$ (meaning it is nontrivial and contains no proper nontrivial normal subgroup of $G/N_{i-1}$). Chief series are related to refinement theorems such as the Schreier refinement theorem and connect to composition factors (though chief factors need not be simple ).

Examples:

• In $S_3$, the only nontrivial proper normal subgroup is $A_3$, so $\{e\}\lhd A_3\lhd S_3$ is a chief series.
• In $A_4$, the Klein four subgroup $V_4$ is normal and there is no nontrivial normal subgroup properly contained in $V_4$, so $\{e\}\lhd V_4\lhd A_4$ is a chief series.
• If $G$ is simple and nontrivial, then $\{e\}\lhd G$ is a chief series (there are no intermediate normal subgroups).