Jordan-Hölder Theorem
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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.
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)