Cohomology and Extensions
If a group acts on a group (on the left, say) by group automorphism
then there is a semidirect product group whose underlying set is the Cartesian product but whose multiplication is twisted by :
for , where denotes the result of acting with on the left on .
If the twist is trivial, then this reduces to just the direct product group construction, whence the name.
There is a projection morphism , . A section of this can be identified with a derivation , i.e. satisfies .
Interior semidirect products
Let be any group. A decomposition of as an internal semidirect product consists of a subgroup and a normal subgroup , such that every element of can be written uniquely in the form , for and .
The internal and external concepts are equivalent. In particular, any (external) semidirect product is an internal semidirect product of the images of and in it.
Right semidirect products
The definitions above are not symmetric in left and right; since the first definition begins with a left action, we may call it a left semidirect product. Then a right semidirect product is given by an action on the right, or internally by the requirement that every element can be written in the form .
However, right and left semidirect products are equivalent. Essentially, this is because any left action defines a right action and vice versa.
Semidirect products of groupoids
It is useful to generalise this to the case is a groupoid. This occurs if for example where is a (left) -space.
So if , then has object set and a morphism is a pair such that in . The composition law is then given again by
if , so that in .
If is a discrete groupoid, and so identified with , then we get which is the action groupoid of the action. In this case the projection is a covering morphism of groupoids, i.e. any has a unique lifting with given initial point. Note that if is a covering map of spaces, then the induced morphism of fundamental groupoids is a covering morphism of groupoids. If is a covering morphism of groupoids, and admits a universal covering map, then there is a topology on such that . In this way, the category of covering maps of is equivalent to the category of covering morphisms of .
The utility of the more general construction is that there is notion of orbit groupoid (identify any and ) and it is a theorem that the orbit groupoid is isomorphic to the quotient groupoid
where is the normal closure in of all elements . Details are in the book reference below (but the conventions are not quite the same).
As split group extensions
Semidirect product groups are precisely the split group extensions of by . See at group extension – split extensions and semidirect product groups.
The automorphisms on the circle group
For the circle group, the automorphism group is
where the nontrivial element in acts on by multiplication with . Write for the automorphism action. The corresponding semidirect product group is the group extension
where the group operation is given by
A general survey is in
Lecture notes include
- Patrick Morandi, Semidirect products (pdf)
Relevant textbooks include
R. Brown, Topology and groupoids, Booksurge 2006.
P. J. Higgins and J. Taylor, The Fundamental Groupoid and Homotopy Crossed Complex of an Orbit Space, in K.H. Kamps et al., ed., Category Theory: Proceedings Gummersbach 1981, Springer LNM 962 (1982) 115–122.