Semidirect product

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