Semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: 
 an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as  splitting extension). Inner semidirect product definitions 

Given a group  with identity element , a subgroup , and a normal subgroup , the following statements are equivalent:
  is the product of subgroups, , and these subgroups have trivial intersection: . For every , there are unique  and  such that . For every , there are unique  and  such that . The composition  of the natural embedding  with the natural projection  is an isomorphism between  and the quotient group . There exists a homomorphism  that is the identity on  and whose kernel is . In other words, there is a split exact sequence
 

 of groups (which is also known as group extension of  by ). If any of these statements holds (and hence all of them hold, by their equivalence), we say  is the semidirect product of  and , written

 or  

or that  splits over ; one also says that  is a semidirect product of  acting on , or even a semidirect product of  and . To avoid ambiguity, it is advisable to specify which is the normal subgroup. If , then there is a group homomorphism  given by , and for , we have . Inner and outer semidirect products 

Let us first consider the inner semidirect product. In this case, for a group , consider its normal subgroup   and the subgroup  (not necessarily normal). Assume that the
conditions on the list above hold. Let  denote the group of all automorphisms of , which is a group under composition. Construct a group homomorphism   defined by conjugation, 
, for all  in  and  in . In this way we can construct a group  with group operation defined as 
 for  in  and  in . The subgroups  and  determine  up to isomorphism, as we will show later. In this way, we can construct the group  from its subgroups. This kind of construction is called an inner semidirect product (also known as internal semidirect product). Let us now consider the outer semidirect product. Given any two groups  and  and a group homomorphism , we can construct a new group , called the outer semidirect product of  and  with respect to , defined as follows:

This defines a group in which the identity element is  and the inverse of the element  is . Pairs  form a normal subgroup isomorphic to , while pairs  form a subgroup isomorphic to . The full group is a semidirect product of those two subgroups in the sense given earlier. Conversely, suppose that we are given a group  with a normal subgroup  and a subgroup , such that every element  of  may be written uniquely in the form  where  lies in  and  lies in . Let  be the homomorphism (written ) given by

for all . Then  is isomorphic to the semidirect product . The isomorphism  is well defined
by  due to the uniqueness of the decomposition . In , we have 

Thus, for  and  we obtain

which proves that  is a homomorphism. Since  is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in . The direct product is a special case of the semidirect product. To see this, let  be the trivial homomorphism (i.e., sending every element of  to the identity automorphism of ) then  is the direct product . A version of the splitting lemma for groups states that a group  is isomorphic to a semidirect product of the two groups  and  if and only if there exists a short exact sequence

and a group homomorphism  such that , the identity map on . In this case,  is given by , where

Examples

Dihedral group 
The dihedral group  with  elements is isomorphic to a semidirect product of the cyclic groups  and . Here, the non-identity element of  acts on  by inverting elements; this is an automorphism since  is abelian. The presentation for this group is:

Cyclic groups 
More generally, a semidirect product of any two cyclic groups  with generator  and  with generator  is given by one extra relation, , with  and  coprime, and ; that is, the presentation:

If  and  are coprime,  is a generator of  and , hence the presentation:

gives a group isomorphic to the previous one. Holomorph of a group 
One canonical example of a group expressed as a semi-direct product is the holomorph of a group. This is defined aswhere  is the automorphism group of a group  and the structure map  comes from the right action of  on . In terms of multiplying elements, this gives the group structure

Fundamental group of the Klein bottle 
The fundamental group of the Klein bottle can be presented in the form

and is therefore a semidirect product of the group of integers, , with . The corresponding homomorphism  is given by . Upper triangular matrices 
The group  of upper triangular matrices with non-zero determinant, that is with non-zero entries on the diagonal, has a decomposition into the semidirect product
 where  is the subgroup of matrices with only 's on the diagonal, which is called the upper unitriangular matrix group, and  is the subgroup of diagonal matrices. The group action of  on  is induced by matrix multiplication. If we set

and 

then their matrix product is

This gives the induced group action 

A matrix in  can be represented by matrices in  and . Hence . Group of isometries on the plane 
The Euclidean group of all rigid motions (isometries) of the plane (maps  such that the Euclidean distance between  and  equals the distance between  and  for all  and  in ) is isomorphic to a semidirect product of the abelian group  (which describes translations) and the group  of orthogonal  matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and , and that the corresponding homomorphism  is given by matrix multiplication: . Orthogonal group O(n) 
The orthogonal group  of all orthogonal real  matrices (intuitively the set of all rotations and reflections of -dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group  (consisting of all orthogonal matrices with determinant , intuitively the rotations of -dimensional space) and . If we represent  as the multiplicative group of matrices , where  is a reflection of -dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant  representing an involution), then  is given by  for all H in  and  in . In the non-trivial case ( is not the identity) this means that  is conjugation of operations by the reflection (in 3-dimensional space a rotation axis and the direction of rotation are replaced by their "mirror image"). Semi-linear transformations 
The group of semilinear transformations on a vector space  over a field , often denoted , is isomorphic to a semidirect product of the linear group  (a normal subgroup of ), and the automorphism group of . Crystallographic groups 
In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic. Non-symmorphic space groups have point groups that are not even contained as subset of the space group, which is responsible for much of the complication in their analysis. Non-examples 
Of course, no simple group can be expressed as a semi-direct product (because they do not have nontrivial normal subgroups), but there are a few common counterexamples of groups containing a non-trivial normal subgroup that nonetheless cannot be expressed as a semi-direct product. Note that although not every group  can be expressed as a split extension of  by , it turns out that such a group can be embedded into the wreath product  by the universal embedding theorem. Z4 
The cyclic group  is not a simple group since it has a subgroup of order 2, namely  is a subgroup and their quotient is , so there's an extensionIf the extension was split, then the group  inwould be isomorphic to . Q8 
The group of the eight quaternions  where  and , is another example of a group which has non-trivial normal subgroups yet is still not split. For example, the subgroup generated by  is isomorphic to  and is normal. It also has a subgroup of order  generated by . This would mean  would have to be a split extension in the following hypothetical exact sequence of groups: , but such an exact sequence does not exist. This can be shown by computing the first group cohomology group of  with coefficients in , so  and noting the two groups in these extensions are  and the dihedral group . But, as neither of these groups is isomorphic with , the quaternion group is not split. This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while  is non-abelian, and noting the only normal subgroups are  and , but  has three subgroups isomorphic to . Properties 
If  is the semidirect product of the normal subgroup  and the subgroup , and both  and  are finite, then the order of  equals the product of the orders of  and . This follows from the fact that  is of the same order as the outer semidirect product of  and , whose underlying set is the Cartesian product . Relation to direct products 
Suppose  is a semidirect product of the normal subgroup  and the subgroup . If  is also normal in , or equivalently, if there exists a homomorphism  that is the identity on  with kernel , then  is the direct product of  and . The direct product of two groups  and  can be thought of as the semidirect product of  and  with respect to  for all  in . Note that in a direct product, the order of the factors is not important, since  is isomorphic to . 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, a semidirect product of two groups is not, in general, unique; if  and  are two groups that both contain isomorphic copies of  as a normal subgroup and  as a subgroup, and both are a semidirect product of  and , then it does not follow that  and  are isomorphic because the semidirect product also depends on the choice of an action of  on . For example, there are four non-isomorphic groups of order 16 that are semidirect products of  and ; in this case,  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: . Existence 

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 of the normal subgroup is coprime to the order of the quotient group. For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct 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. Generalizations 

Within group theory, the construction of semidirect products can be pushed much further. The Zappa–Szep 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 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 semidirect sum of Lie algebras. For geometry, there is also a crossed product for 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; c.f. noncommutative geometry. The semidirect product is a special case of the Grothendieck construction in category theory. Specifically, an action of  on  (respecting the group, or even just monoid structure) is the same thing as a functor

from the groupoid  associated to H (having a single object *, whose endomorphisms are H) to the category of categories such that the unique object in  is mapped to . The Grothendieck construction of this functor is equivalent to , the (groupoid associated to) semidirect product. Groupoids 

Another generalization is for groupoids. This occurs in topology because if a group  acts on a space  it also acts on the fundamental groupoid  of the space. The semidirect product  is then relevant to finding the fundamental groupoid of the orbit space . For full  details see Chapter 11 of the book referenced below, and also some details in semidirect product in 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. Notation 

Usually the semidirect product of a group  acting on a group  (in most cases by conjugation as subgroups of a common group) is denoted by  or . However, some sources may use this symbol with the opposite meaning. In case the action  should be made explicit, one also writes . One way of thinking about the  symbol is as a combination of the symbol for normal subgroup () and the symbol for the product (). Barry Simon, in his book on group representation theory, employs the unusual notation  for the semidirect product. Unicode lists four variants:

{| class="wikitable"
! ! ! Value  ! ! MathML ! ! Unicode description
|-
| ⋉ || U+22C9 || ltimes || LEFT NORMAL FACTOR SEMIDIRECT PRODUCT
|-
| ⋊ || U+22CA || rtimes || RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT
|-
| ⋋ || U+22CB || lthree || LEFT SEMIDIRECT PRODUCT
|-
| ⋌ || U+22CC || rthree || RIGHT SEMIDIRECT PRODUCT
|}

Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice. In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters. With the AMS symbols package loaded, \leftthreetimes produces ⋋ and \rightthreetimes produces ⋌. See also 
Affine Lie algebra
Grothendieck construction, a categorical construction that generalizes the semidirect product
Holomorph
Lie algebra semidirect sum
Subdirect product
Wreath product
Zappa–Szép product
Crossed product

Notes

References

 
 

Group products