Editing Semidirect product (section)

== Properties ==
If {{math|''G''}} is the semidirect product of the normal subgroup {{math|''N''}} and the subgroup {{math|''H''}}, and both {{math|''N''}} and {{math|''H''}} are finite, then the [[order of a group|order]] of {{math|''G''}} equals the product of the orders of {{math|''N''}} and {{math|''H''}}. This follows from the fact that {{math|''G''}} is of the same order as the outer semidirect product of {{math|''N''}} and {{math|''H''}}, whose underlying set is the [[Cartesian product]] {{math|''N'' × ''H''}}.

=== Relation to direct products ===
Suppose {{math|''G''}} is a semidirect product of the normal subgroup {{math|''N''}} and the subgroup {{math|''H''}}. If {{math|''H''}} is also normal in {{math|''G''}}, or equivalently, if there exists a homomorphism {{math|''G'' → ''N''}} that is the identity on {{math|''N''}} with kernel {{math|''H''}}, then {{math|''G''}} is the [[direct product of groups|direct product]] of {{math|''N''}} and {{math|''H''}}.

The direct product of two groups {{math|''N''}} and {{math|''H''}} can be thought of as the semidirect product of {{math|''N''}} and {{math|''H''}} with respect to {{math|''φ''(''h'') {{=}} id{{sub|''N''}}}} for all {{math|''h''}} in {{math|''H''}}.

Note that in a direct product, the order of the factors is not important, since {{math|''N'' × ''H''}} is isomorphic to {{math|''H'' × ''N''}}. This is not the case for semidirect products, as the two factors play different roles.

Furthermore, the result of a (proper) semidirect product by means of a non-trivial homomorphism is never an [[abelian group]], even if the factor groups are abelian.

=== Non-uniqueness of semidirect products (and further examples) ===
As opposed to the case with the [[direct product of groups|direct product]], a semidirect product of two groups is not, in general, unique; if {{math|''G''}} and {{math|''G′''}} are two groups that both contain isomorphic copies of {{math|''N''}} as a normal subgroup and {{math|''H''}} as a subgroup, and both are a semidirect product of {{math|''N''}} and {{math|''H''}}, then it does ''not'' follow that {{math|''G''}} and {{math|''G′''}} are [[group isomorphism|isomorphic]] because the semidirect product also depends on the choice of an action of {{math|''H''}} on {{math|''N''}}.

For example, there are four non-isomorphic groups of order 16 that are semidirect products of {{math|C{{sub|8}}}} and {{math|C{{sub|2}}}}; in this case, {{math|C{{sub|8}}}} is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups:
* the dihedral group of order 16
* the [[quasidihedral group]] of order 16
* the [[Iwasawa group]] of order 16

If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 (the only one containing six elements of order 4 and six elements of order 6) that can be expressed as semidirect product in the following ways: {{math|(D{{sub|8}} ⋉ C{{sub|3}}) ≅ (C{{sub|2}} ⋉ [[Dicyclic group|Q{{sub|12}}]]) ≅ (C{{sub|2}} ⋉ D{{sub|12}}) ≅ (D{{sub|6}} ⋉ [[Klein four-group|V]])}}.<ref name="Rose2009">{{cite book|author=H.E. Rose|title=A Course on Finite Groups|year=2009|publisher=Springer Science & Business Media|isbn=978-1-84882-889-6|page=183}} Note that Rose uses the opposite notation convention than the one adopted on this page (p. 152).</ref>

=== Existence ===
{{main|Schur–Zassenhaus theorem}}
In general, there is no known characterization (i.e., a necessary and sufficient condition) for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. For finite groups, the [[Schur–Zassenhaus theorem]] guarantees existence of a semidirect product when the [[order (group theory)|order]] of the normal subgroup is [[coprime]] to the order of the [[quotient group]].

For example, the Schur–Zassenhaus theorem implies the existence of a semidirect product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.