Jordan-Hölder Theorem

Theorem

Let $G$ be a finite group.

Let $\HH_1$ and $\HH_2$ be two composition series for $G$.

Then:

$\HH_1$ and $\HH_2$ have the same length
Corresponding factors of $\HH_1$ and $\HH_2$ are isomorphic.


Proof

By the Schreier-Zassenhaus Theorem, two normal series have refinements of equal length whose factors are isomorphic.

But from the definition of composition series, $\HH_1$ and $\HH_2$ have no proper refinements.

Hence any such refinements must be identical to $\HH_1$ and $\HH_2$ themselves.

$\blacksquare$


Source of Name

This entry was named for Marie Ennemond Camille Jordan and Otto Ludwig Hölder.


Historical Note

The was proved by Camille Jordan in $1869$, and then independently rediscovered by Otto Ludwig Hölder in $1889$.


Sources

  • 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 73$. The Jordan-Hölder THeorem
  • 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.9$
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Jordan-Hölder theorem (C. Jordan 1869, O. Hölder 1889)
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