Jordan–Hölder Theorem (Groups)

Jordan–Hölder Theorem (Groups). Let $G$ be a group that admits a composition series , i.e. a finite chain

such that each $G_{i+1}\trianglelefteq G_i$ and each factor $G_i/G_{i+1}$ is a simple group . Then for any two composition series of $G$,

• the lengths are equal, and
• the multisets of factor groups $\{G_i/G_{i+1}\}$ agree up to isomorphism and permutation.

Jordan–Hölder gives a well-defined notion of the “composition factors” of a group (up to order and isomorphism). The standard proof combines the Schreier refinement theorem with the fact that a simple factor admits no nontrivial refinement.

Proof sketch. Apply Schreier refinement to two composition series to obtain equivalent refinements. Since factors in a composition series are simple, any refinement must repeat the same factors (there is no proper intermediate normal subgroup), forcing the original series to have the same factors up to order.