nLab Jordan-Hölder theorem

Idea

The Jordan-Hölder theorem says that every composition series of a given group, and every Jordan-Hölder sequence on a given object in an abelian category, has the same length, and the same simple factors, up to permutation. In particular says that the length of an object in an abelian category is well defined.

More generally, a form of the theorem holds in any homological category.

Proof

This is a proof of the classical version of the theorem.

Theorem

Every composition series of a given finite group has the same length and the same simple factors up to permutation.

Proof

Let $G$ be a finite group with two composition series $G = G_0 \rhd ... \rhd G_n$ (a) and $G = G_0\prime \rhd ... \rhd G_m\prime$ (b) where $G_n = G_m\prime = 1$ (the trivial group) and $n$, $m$ are nonzero natural numbers. Proceed by induction on $|G|$. The base case is trivial. In the inductive case:

If $G_1 = G_1\prime$ (case A), then we have two composition series for the same group of order less than $|G|$. By induction, $n = m$ and we have a permutation relating the factor groups, which can be easily extended (as $G_0/G_1 = G_0\prime/G_1\prime$), so this case is done.

If $G_1 \neq G_1\prime$ (case B), then we have $G_1 \lhd G$ and $G_1\prime \lhd G$, so $G_1 G_1\prime = \{ g g\prime | g \in G_1, g\prime \in G_1\prime \} \lhd G$. As both $G_1$ and $G_1\prime$ are both maximal normal in $G$ and $G_1 G_1\prime$ contains both, $G_1 G_1\prime = G$. Thus, by the second isomorphism theorem? for groups, $G/G_1 \cong G_1\prime/(G_1 \cap G_1\prime)$ and $G/G_1\prime \cong G_1/(G_1 \cap G_1\prime)$. Let $H \coloneqq G_1 \cap G_1\prime$; due to the previous isomorphisms $H$ is maximal normal in both $G_1$ and $G_1\prime$.

Now, let $H = H_2 \rhd ... \rhd H_l$ be a composition series for $H$. Thus we have two new composition series $G = G_0 \rhd G_1 \rhd H_2 \rhd ... \rhd H_l = 1$ (c) and $G = G_0\prime \rhd G_1\prime \rhd H_2 \rhd ... \rhd H_l = 1$ (d). By case A we see that $n = l$ and that there are isomorphisms relating (a) and (c). Similarly, by case A we also see that $m = l$ and that there are isomorphisms relating (b) and (d). These, as well as the isomorphisms $G_0/G_1 \cong G_1\prime/H_2$ and $G_0\prime/G_1\prime \cong G_1/H_2$, allow us to connect (a) and (b) through (c) and (d).

Relation to FTA

The theorem can be seen as a generalization of the fundamental theorem of arithmetic. Using cyclic groups to represent positive natural numbers, simple cyclic groups correspond to prime numbers and composition series correspond to prime factorizations.

References

• David Speyer, The Jordan-Hölder theorem (pdf)