A theorem in
group theory, this states that the
composition factors of a
group are unique, up to the order you write them in.
In more technical terms, if G is a
group and
G=G
1>G
2>...>G
n>G
n+1={1}
is a
composition series for G, then the
composition factors G
i/G
i+1 are unique to within rearrangement.
ie. if
G=H1>H2>...>Hk>Hk+1={1}
is another composition series for G, then n=k and there is a permutation f in Sn such that Gi/Gi+1 is isomorphic to Hf(i)/Hf(i)+1.
The proof of the Jordan-Hölder theorem is by induction on the size of the group, and is an application of the second and third isomophism theorems.