Subnormal series

Definition

Let $G$ be a group. A subnormal series of $G$ is a finite sequence of subgroups $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$, where each $G_{i-1}$ is a normal subgroup of $G_i$. The quotient groups $G_i / G_{i-1}$ are the factors of the series, and $n$ is its length.

Remarks

The subgroup $G_{i-1}$ needs to be normal only in $G_i$, not in $G$. When each $G_i$ is normal in $G$, the series is called a normal series. For example, $\{e\} \trianglelefteq V \trianglelefteq A_4 \trianglelefteq S_4$ is a normal series of $S_4$, where $V$ is the Klein four-group in $S_4$.

Explanations

L1 human

A subnormal series cuts a group into layers. Each layer is a quotient group, and you can study the layers one at a time. Good series have simple layers, like composition series, or abelian layers, like the series of a solvable group.