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=G1>G2>...>Gn>Gn+1={1} is a composition series for G, then the composition factors Gi/Gi+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.