Schur–Zassenhaus theorem
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The Schur–Zassenhaus theorem is a theorem in group theory which states that if G is a finite group, and N is a normal subgroup whose order is coprime to the order of the quotient group G / N, then G is a semidirect product of N and G / N.
An alternative statement of the theorem is that any normal Hall subgroup of a finite group G has a complement in G.
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.
[edit] References
- Rotman, Joseph J. (1995). An Introduction to the Theory of Groups. New York: Springer–Verlag. ISBN 978-0-387-94285-8.
- David S. Dummit & Richard M. Foote (2003). Abstract Algebra. Wiley. ISBN 978-0-471-43334-7.
- J. S. Milne (2003). Group Theory. Lecture notes.
