Composition series

A composition series is a way of describing a group.

Definition

A composition series of a group $G$ with idenitity $e$ is a finite sequence $(G_i)_{0\le i \le n}$ of subgroups of $G$ such that $G_0 =G$, $G_n= \{e\}$, and for each integer $i \in [0,n-1]$, $G_{i+1}$ is a normal subgroup of $G_i$.

The quotient groups $G_i/G_{i+1}$ are called the quotients of the series. We call a composition series $\Sigma_1$ finer than a composition series $\Sigma_2$ if the terms of $\Sigma_2$ are taken from the terms of $\Sigma_1$. Note, however, that in general, a composition series with some terms omitted is no longer a composition series, since in general if $K$ is a normal subgroup of $H$ and $H$ is a normal subgroup of $G$, then $K$ is not necessarily a normal subgroup of $G$.

Two composition series $(G_i)_{0 \le i \le n}$ and $(H_i)_{0\le i \le m}$ (of not necessarily identical groups $G$ and $H$) are considered equivalent if $m=n$, and there is a permutation $\sigma$ of the integers in $[0,n-1]$ such that $G_i/G_{i+1}$ and $H_{\sigma(i)}/H_{\sigma(i)+1}$ are isomorphic for all integers $i \in [0,n-1]$.

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