Schur-Zassenhaus theorem
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The Schur-Zassenhaus theorem is a theorem in group theory which states that if G is a group, and N is a normal subgroup whose order is coprime to the quotient group G / N, then G is a semidirect product of N and G / N.
It is clear that if we do not impose the coprime condition, the theorem is not true: consider for example the cyclic group C4 and its normal subgroup C2. Then if C4 were a semidirect product of C2 and 
then C4 would have to contain two elements of order 2, but it only contains one.
The Schur-Zassenhaus theorem at least partially answers the question: "In a composition series, how can we classify groups with a certain set of composition factors?" The other part, which is where the composition factors do not have coprime orders, is tackled in extension theory.
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