Jordan–Hölder theorem

Statement

Let $G$ be a group and let $\{e\} = G_0 \trianglelefteq \cdots \trianglelefteq G_n = G$ and $\{e\} = H_0 \trianglelefteq \cdots \trianglelefteq H_m = G$ be two composition series of $G$. Then $n = m$, and there is a permutation $\sigma$ of $\{1, \dots, n\}$ with $G_i / G_{i-1} \cong H_{\sigma(i)} / H_{\sigma(i) - 1}$ for each $i$.

Remarks

So the composition factors and the composition length of $G$ are well defined. For $G = \mathbb{Z}/n\mathbb{Z}$, the theorem gives the uniqueness of the prime factorisation of $n$. One proof uses the Schreier refinement theorem, which says that two subnormal series have equivalent refinements.

History

The theorem is named after Camille Jordan and Otto Hölder.

Explanations

L1 human

A group can be cut into simple layers in many ways. The theorem says that the layers that you get are always the same, up to order. This is the group version of the uniqueness of prime factorisation.