Editing Semidirect product (section)

=== 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>