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=H_{1}>H_{2}>...>H_{k}>H_{k+1}={1}

is another composition series for G, then n=k and there is a permutation f in S_{n} such that G_{i}/G_{i+1} is isomorphic to H_{f(i)}/H_{f(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.