Jordan–Hölder Theorem

Composition series exist and their simple factors are unique.

The idea

A finite group can be taken apart into simple pieces, and the Jordan–Hölder theorem says the pieces do not depend on how we take it apart.

A group is simple when its only normal subgroups are the trivial group and itself, so it admits no further splitting into a normal subgroup and a quotient.

Definition.

A composition series for a group $G$ is a chain $1 = N_0 \trianglelefteq N_1 \trianglelefteq \cdots \trianglelefteq N_k = G$ in which each term is normal in the next and every quotient $N_{i+1}/N_i$ is simple. The quotients $N_{i+1}/N_i$ are the composition factors of the series, and $k$ is its length.

Theorem (Jordan–Hölder Theorem).

Every finite group has a composition series, and any two composition series of the same group have the same length and the same composition factors, up to reordering and isomorphism.

A group typically has many composition series, passing through different subgroups; the theorem makes the length and the factors invariants of $G$ itself. The figure shows the smallest case: two series climb from $1$ to the same $G$ through different subgroups $A$ and $B$, yet both meet one factor of order $p$ and one of order $q$. In particular, two groups with different factor lists are not isomorphic. The converse fails: the cyclic group of order $6$ and the symmetric group $S_3$ both have factors $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$, yet only the first is abelian. The factors determine the pieces of $G$, not the way the pieces are assembled.

Ways to work on it

• Walkthrough. Read the composition factors off a series and see what Jordan–Hölder pins down.
• Practice. Count the composition factors of a cyclic group.
• Hardest. Refine a series of a nonabelian group down to its simple factors.

Not sure where to start? Take the ten-question placement test.