The Jordan-Hölder Theorem

# The Jordan-Hölder Theorem

Recall from the Composition Series in a Group page that if $G$ is a group then a composition series for $G$ is a chain of successive subgroups:

(1)
\begin{align} \quad \{ e \} = G_0 \leq G_1 \leq ... \leq G_n = G \end{align}

With the following properties:

• 1) $G_i$ is a normal subgroup of $G_{i+1}$ for all $0 \leq i \leq n - 1$.
• 2) $G_{i+1}/G_i$ is a simple group for all $0 \leq i \leq n - 1$.

We said that the composition series was the positive integer $n$, and that the composition factors of the composition series were the quotient groups $G_{i+1}/G_i$. As we saw on the page referenced above - unfortunately, composition series are not necessarily unique.

The first result we will look at tells us that every finite group has a composition series.

 Lemma 2: Let $G$ be a group. If $G$ is a finite group then $G$ has a composition series.
• Proof: We prove the result by induction. Suppose that every group of order less than $|G|$ has a composition series.
• Now if $G$ is a simple group then $1 \triangleleft G$ is a composition series. If $G$ is not a simple group then there exists a nontrivial proper normal subgroup of $G$. Since $G$ is a finite group, a maximal nontrivial proper normal subgroup exists. Denote this subgroup by $H$. Since $H$ is a proper subgroup of $G$ we have that $|H| < |G|$. By the induction hypothesis, $H$ has a composition series:
(2)
\begin{align} \quad \{1 \} = H_0 \leq H_1 \leq ... \leq H_k = H \end{align}
• But then $H \triangleleft G$ by the maximality of $H$. So:
(3)
\begin{align} \quad \{ 1 \} = H_0 \leq H_1 \leq ... \leq H_k = H \leq G \end{align}
• The above chain of subgroups is a composition series of $G$. $\blacksquare$

We now state a very important theorem which tells us that any two composition series for a group $G$ are the same in the sense that their sets of composition factors are isomorphically the same.

 Theorem 2 (The Jordan-Hölder Theorem): Let $G$ be a group. If $G$ has two composition series, then there exists a bijection from the set of composition factors from the first composition series to the set of composition factors from the second composition series, such that each composition factor is isomorphic to its image under the bijection.