Editing Semidirect product (section)

== Generalizations ==

Within group theory, the construction of semidirect products can be pushed much further. The [[Zappa–Szép product]] of groups is a generalization that, in its internal version, does not assume that either subgroup is normal. 

There is also a construction in [[ring theory]], the [[crossed product|crossed product of rings]].  This is constructed in the natural way from the [[group ring]] for a semidirect product of groups.  The ring-theoretic approach can be further generalized to the [[Lie algebra extension#By semidirect sum|semidirect sum of Lie algebras]].  

For geometry, there is also a crossed product for [[Group action (mathematics)|group actions]] on a [[topological space]]; unfortunately, it is in general non-commutative even if the group is abelian.  In this context, the semidirect product is the ''space of orbits'' of the group action.  The latter approach has been championed by [[Alain Connes]] as a substitute for approaches by conventional topological techniques; cf. [[noncommutative geometry]].  

The semidirect product is a special case of the [[Grothendieck construction]] in [[category theory]]. Specifically, an action of <math>H</math> on <math>N</math> (respecting the group, or even just monoid structure) is the same thing as a [[functor]]
: <math>F : BH \to Cat</math>
from the [[groupoid]] <math>BH</math> associated to ''H'' (having a single object *, whose endomorphisms are ''H'') to the category of categories such that the unique object in <math>BH</math> is mapped to <math>BN</math>. The Grothendieck construction of this functor is equivalent to <math>B(H \rtimes N)</math>, the (groupoid associated to) semidirect product.<ref>{{harvtxt|Barr|Wells|2012|loc=§12.2}}</ref>

=== Groupoids ===

Another generalization is for groupoids. This occurs in topology because if a group {{math|''G''}} acts on a space {{math|''X''}} it also acts on the [[fundamental groupoid]] {{math|''π''{{sub|1}}(''X'')}} of the space.  The semidirect product {{math|''π''{{sub|1}}(''X'') ⋊ ''G''}} is then relevant to finding the fundamental groupoid of the [[orbit space]] {{math|''X/G''}}. For full  details see Chapter 11 of the book referenced below, and also some details in semidirect product<ref>{{cite web| url = http://ncatlab.org/nlab/show/semidirect+product| title = Ncatlab.org}}</ref> in [[nLab|ncatlab]].

=== Abelian categories ===
Non-trivial semidirect products do ''not'' arise in [[abelian categories]], such as the [[category of modules]]. In this case, the [[splitting lemma]] shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.