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 (aka split[ting] extension) of $N$ and $G/N$ .

An alternative statement of the theorem is that any normal Hall subgroup $N$ of a finite group $G$ has a complement in $G$ . In some textbooks (e.g. Isaac or Rose), the Schur–Zassenhaus theorem is stated with the additional conclusion that all complements of $N$ in G are conjugate. All known proofs of the conjugacy conclusion require the use of the (more recent) Feit-Thompson theorem.

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.

According to Rotman, Issai Schur (1904) proved the theorem for the special case when the complement of $N$ is cyclic and Hans Zassenhaus (1937) extended it for arbitrary complements. Kurzweil and Stellmacher cite a 1907 publication in of Schur in Crelle's Journal for his result. Humphreys cites a similarly titled 1904 paper of Schur (in the same journal) for the first results in central extensions, which are motivated by failure of the semidirect construction.

Examples and non-examples

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. Another way to explain this impossibility of splitting $C_4$ (i.e. expressing it as a semidirect product) is to observe that the automorphisms of $C_2$ are the trivial group, so the only possible [semi]direct product of $C_2$ with itself is a direct product (which gives rise to the Klein four-group, a group that is non-isomorphic with $C_4$ ).

An example where the Schur–Zassenhaus theorem does apply is the symmetric group on 3 symbols, $S_3$ , which has a normal subgroup of order 3 (isomorphic with $C_3$ ) which in turn has index 2 in $S_3$ (in agreement with the theorem of Lagrange), so $S_3 / C_3 \cong C_2$ . Since 2 and 3 are relatively prime, the Schur–Zassenhaus theorem applies and $S_3 \cong C_3 \rtimes C_2$ . Note that the automorphism group of $C_3$ is $C_2$ and the automorphism of $C_3$ used in the semidirect product that gives rise to $S_3$ is the non-trivial automorphism that permutes the two non-identity elements of $C_3$ . Furthermore, the three subgroups of order 2 in $S_3$ (any of which can serve as a complement to $C_3$ in $S_3$ ) are conjugate to each other.

The non-triviality of the (additional) conjugacy conclusion can be illustrated with the Klein four-group $V$ as the non-example. Any of the three proper subgroups of $V$ (all of which have order 2) is normal in $V$ ; fixing one of these subgroups, any of the other two remaining (proper) subgroups complements it in $V$ , but none of these three subgroups of $V$ is a conjugate of any other one, because $V$ is Abelian.

The converse of the Schur–Zassenhaus theorem does not hold, i.e. there are semidirect products where the coprimality of index and order doesn't hold. For example, the dihedral group of order 8 is isomorphic with $C_4 \rtimes C_2$ . It is also worth noting that there exists another group of order 8, the quaternion group that has normal subgroups of order 4 and 2 but is not a [semi]direct product. Schur's paper(s) at the beginning of the 20th century introduced the notion of central extension to address examples such as such $C_4$ and the quaternions.

References

Joseph J. Rotman (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.
John S. Rose (1978). A Course on Group Theory. Cambridge University Press. ISBN 0-521-21409-2.
I. Martin Isaacs (2008). Finite Group Theory. American Mathematical Society. ISBN 978-0-8218-4344-4.
Hans Kurzweil & Bernd Stellmacher (2004). The Theory of Finite Groups: An Introduction. Springer-Verlag. ISBN 0-387-40510-0.
James E. Humphreys (1996). A Course in Group Theory. Oxford University Press. ISBN 0-19-853459-0.
J. Schur: Über die Darstellung der endlichen Gruppen durch gebrochen lineare Substitutionen. Journal für die reine und angewandte Mathematik 127 (1904): 20-50.
J. Schur: Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine u. angew. Math. 132 (1907), 85-137.
H. Zassenhaus: Lehrbuch der Gruppentheorie. Leipzig 1937.