Composition series

From AoPSWiki

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 $G$ .