Composition series

Let $G$ be a group . A composition series for $G$ is a finite chain of subgroups $\{e\}=G_0\lhd G_1\lhd \cdots \lhd G_n=G$ such that:

1. each $G_{i-1}$ is normal in $G_i$ (so this is a subnormal series ), and
2. each factor quotient group $G_i/G_{i-1}$ is a simple group .

The groups $G_i/G_{i-1}$ are called the composition factors of the series. A major structural result, the Jordan–Hölder theorem , says that although a composition series is not unique, the multiset of isomorphism types of composition factors is unique up to permutation.

Examples:

• For $S_3$, the chain $\{e\}\lhd A_3\lhd S_3$ is a composition series; the factors are $A_3/\{e\}\cong C_3$ and $S_3/A_3\cong C_2$.
• If $G\cong \mathbb{Z}/p^n\mathbb{Z}$, then the chain $p^n\mathbb{Z}/p^n\mathbb{Z}\lhd p^{n-1}\mathbb{Z}/p^n\mathbb{Z}\lhd\cdots\lhd p\mathbb{Z}/p^n\mathbb{Z}\lhd \mathbb{Z}/p^n\mathbb{Z}$ yields a composition series with all factors cyclic of order $p$.
• If $G$ is simple and nontrivial, then $\{e\}\lhd G$ is a composition series of length $1$.