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Composition series

Definition

Given a finite group $G$, a composition series for $G$ is a sequence of groups $\{e\} \triangleleft G_1 \triangleleft ... \triangleleft G_{n-1} \triangleleft G_n = G$ where each composition factor $G_{i+1}/G_i$ is simple.

Properties

Jordan-Hölder theorem: All composition series of $G$ have the same length and composition factors, including multiplicity.

all these pages adapted with probably insufficient credit from my university's lecture notes