Schur–Zassenhaus theorem

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 $C\_4$ and its normal subgroup $C\_2$. Then if $C\_4$ were a semidirect product of $C\_2$ and $C\_4\; /\; C\_2\; cong\; C\_2$ then $C\_4$ 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.

References

*cite book | author=Rotman, Joseph J. | title=An Introduction to the Theory of Groups | location=New York | publisher=Springer–Verlag | year=1995 | id=ISBN 978-0-387-94285-8

*cite book | author=David S. Dummit & Richard M. Foote | title=Abstract Algebra | publisher=Wiley | year=2003 | id=ISBN 978-0-471-43334-7

* J. S. Milne (2003). [http://www.jmilne.org/math/CourseNotes/math594g.html Group Theory] . Lecture notes.