Jordan–Hölder Theorem (Groups)

Any two composition series have the same length and the same composition factors up to order

Jordan–Hölder Theorem (Groups)

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

G = G_0 \triangleright G_1 \triangleright \cdots \triangleright G_n = \{e\}

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.