Schur–Zassenhaus theorem

The theorem of Schur - Zassenhaus is a mathematical theorem in group theory. Named after Hans Julius Zassenhaus Isay Schur and sentence reads:

• For a finite group and a normal subgroup with a subgroup exists with and. The group is therefore the semidirect product of and.

The subgroup in the above sentence is in general not uniquely determined, but one can show that any two such subgroups are conjugate.

Examples

• The cyclic group has a normal subgroup. Since the numbers and are relatively prime, the set of Schur - Zassenhaus can be applied. is apparently the only subgroup that satisfies the statement of the theorem. Since the group is abelian, the semi- direct product is even right in this case.

• The symmetric group has the normal subgroup. Paths and the set of Schur - Zassenhaus can be applied, apparently met the three subgroups the statement of the theorem.

• The cyclic group has a normal subgroup. Here are not relatively prime, which is why the sentence is not applicable. Indeed, there is no subgroup that satisfies the statement of the theorem, for such would have an element of order 2, but the only element of order 2, and which has already run. This example shows that the coprimality of and in the above theorem can not be omitted.

• Is any group, as the example shows that the coprime condition is not necessary for the existence of a representation as a semidirect, and even direct, product.