Semidirect product - Citizendia

category of groups
Basic notions in group theory
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

In mathematics, especially in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. A semidirect product is a generalization of a direct product. In Mathematics, one can often define a direct product of objectsalready known giving a new one A semidirect product is a cartesian product as a set, but with a particular multiplication operation. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

1 Some equivalent definitions
2 Elementary facts and caveats
3 Semidirect products and group homomorphisms
4 Examples
5 Relation to direct products
6 Generalizations
7 Notation
8 See also

Some equivalent definitions

Let G be a group, N a normal subgroup of G (i. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. e. , N ◁ G) and H a subgroup of G. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of The following statements are equivalent:

G = NH and N ∩ H = {e} (with e being the identity element of G)
G = HN and N ∩ H = {e}
Every element of G can be written as a unique product of an element of N and an element of H
Every element of G can be written as a unique product of an element of H and an element of N
The natural embedding H → G, composed with the natural projection G → G / N, yields an isomorphism between H and the quotient group G / N
There exists a homomorphism G → H which is the identity on H and whose kernel is N

If one (and therefore all) of these statements hold, we say that G is a semidirect product of N and H, or that G splits over N. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism

Elementary facts and caveats

If G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is

Note that, as opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if G and G' are two groups which both contain N as a normal subgroup and H as a subgroup, and both are a semidirect product of N and H, then it does not follow that G and G' are isomorphic. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in This remark leads to an extension problem, of describing the possibilities.

Semidirect products and group homomorphisms

Let G be a semidirect product of N and H. Let Aut(N) denote the group of all automorphisms of N. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself The map φ : H → Aut(N) defined by φ(h) = φ_h, where φ_h(n) = hnh^-1 for all h in H and n in N, is a group homomorphism. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function Together N, H and φ determine G up to isomorphism, as we show now. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose

Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : H → Aut(N), the new group $N\rtimes_{\varphi}H$ (or simply N ×_φ H) is called the semidirect product of N and H with respect to φ, defined as follows. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function As a set, $N\rtimes_{\varphi}H$ is defined as the cartesian product N × H. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. Multiplication of elements in the cartesian product is determined by the homomorphism φ, with the operation * defined by

$(n_1, h_1)*(n_2, h_2) = (n_1\varphi_{h_1}(n_2), h_1h_2)$

for all n₁, n₂ in N and h₁, h₂ in H. This is a group in which the identity element is (e_N, e_H) and the inverse of the element (n, h) is (φ_h^–1(n^–1), h^–1). Pairs (n,e_H) form a normal subgroup isomorphic to N, while pairs (e_N, h) form a subgroup isomorphic to H. The full group is a semidirect product of those two subgroups in the sense given above.

Conversely, suppose that we are given a group G with a normal subgroup N, a subgroup H, and such that every element g of G may be written uniquely in the form g=nh where n lies in N and h lies in H. Let φ : H→Aut(N) be the homomorphism given by φ(h) = φ_h, where

$\varphi_h(n) = hnh^{-1}$

for all n in N and h in H. Then G is isomorphic to the semidirect product $N\rtimes_{\phi}H$ ; the isomorphism sends the product nh to the tuple (n,h). In G, we have the multiplication rule

$(n_1h_1)(n_2h_2) = (n_1(h_1n_2h_1^{-1}))(h_1h_2).$

A version of the splitting lemma for groups states that a group G is isomorphic to a semidirect product of the two groups N and H if and only if there exists a short exact sequence

$0\longrightarrow N \longrightarrow^{\!\!\!\!\!\!\!\!\!\beta}\ \, G \longrightarrow^{\!\!\!\!\!\!\!\!\!\alpha}\ \, H \longrightarrow 0$

and a group homomorphism γ : H → G such that $\alpha \circ \gamma = id_H$ , the identity map on H. In Mathematics, and more specifically in Homological algebra, the splitting lemma states that in any Abelian category, the following statements for In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group This article is about the Identity Map software design pattern In this case, φ : H → Aut(N) is given by φ(h) = φ_h, where

φ h (n) = β - 1 (γ(h)β(n)γ(h - 1)).

If φ is the trivial homomorphism, sending every element of H to the identity automorphism of N, then $N\rtimes_{\phi}H$ is the direct product $N \times H$ .

Examples

The dihedral group D_n with 2n elements is isomorphic to a semidirect product of the cyclic groups C_n and C₂. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Here, the non-identity element of C₂ acts on C_n by inverting elements; this is an automorphism since C_n is abelian. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the The presentation for this group is:

$\langle a,\;b \mid a^2 = e,\; b^n = e,\; aba^{-1}=b^{-1}\rangle$ . In Mathematics, one method of defining a group is by a presentation.

More generally, a semidirect product of any two cyclic groups $C_m\;$ with generator $a\;$ and $C_n\;$ with generator $b\;$ is given by a single relation $aba^{-1}=b^k\;$ with $k\;$ and $n\;$ coprime, i. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than e. the presentation:

$\langle a,\;b \mid a^m=e,\;b^n = e,\;aba^{-1}=b^k\;\rangle$ .

If $r\;$ and $m\;$ are coprime, $a^r\;$ is a generator of $C_m\;$ and $a^rba^{-r}=b^{k^r}\;$ , hence the presentation:

$\langle a,\;b \mid a^m=e,\;b^n = e,\;aba^{-1}=b^{k^{r}}\;\rangle$

gives a group isomorphic to the previous one.

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

$\langle a,\;b \mid aba^{-1}=b^{-1}\;\rangle$

and is therefore a semidirect product of the group of integers, $\mathbb{Z}$ , with itself. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, the Klein bottle is a certain non- orientable Surface, i

The Euclidean group of all rigid motions ( isometries) of the plane (maps f : R² → R² such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in R²) is isomorphic to a semidirect product of the abelian group R² (which describes translations) and the group O(2) of orthogonal 2×2 matrices (which describes rotations and reflections which keep the origin fixed). In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional For the Mechanical engineering and Architecture usage see Isometric projection. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i n is a translation, h a rotation or reflection. Applying a translation and then a rotation or reflection corresponds to 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). In a group, the conjugate by g of h is ghg -1 Translation If h is a translation then its conjugate by Every orthogonal matrix acts as an automorphism on R² by matrix multiplication. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

The orthogonal group O(n) of all orthogonal real n×n matrices (intuitively the set of all rotations and reflections of n-dimensional space which keep the origin fixed) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of n-dimensional space) and C₂. In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n If we represent C₂ as the multiplicative group of matrices {I, R}, where R is a reflection of n dimensional space which keeps the origin fixed (i. e. an orthogonal matrix with determinant –1 representing an involution), then φ : C₂ → Aut(SO(n)) is given by φ(H)(N) = H N H^–1 for all H in C₂ and N in SO(n). In the non-trivial case ( H is not the identity) this means that φ(H) is conjugation of operations by the reflection (a rotation axis and the direction of rotation are replaced by their "mirror image").

Relation to direct products

Suppose G is a semidirect product of the normal subgroup N and the subgroup H. If H is also normal in G, or equivalently, if there exists a homomorphism G → N which is the identity on N, then G is the direct product of N and H. In Mathematics, one can often define a direct product of objectsalready known giving a new one

The direct product of two groups N and H can be thought of as the outer semidirect product of N and H with respect to φ(h) = id_N for all h in H.

Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles.

Generalizations

The construction of semidirect products can be pushed much further. The Zappa-Szep product of groups is a generalization which, in its internal version, does not assume that either subgroup is normal. In Mathematics, especially Group theory, the Zappa-Szep product (also known as the knit product) describes a way in which a group can be constructed There is also a construction in ring theory, the crossed product of rings. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those This is seen naturally as soon as one constructs a group ring for a semidirect product of groups. In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively There is also the semidirect sum of Lie algebras. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Given a group action on a topological space, there is a corresponding crossed product which will in general be non-commutative even if the group is abelian. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This kind of ring (see crossed product for a related construction) can play the role of the space of orbits of the group action, in cases where that space cannot be approached by conventional topological techniques - for example in the work of Alain Connes (cf. In Mathematics, and more specifically in the theory of Von Neumann algebras a crossed product is a basic method of constructing a new von Neumann algebra from a von Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University noncommutative geometry). Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the

There are also far-reaching generalisations in category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets They show how to construct fibred categories from indexed categories. Fibred categories are abstract entities in Mathematics used to provide a general framework for Descent theory. This is an abstract form of the outer semidirect product construction.

Notation

Sources differ in their notation for the semidirect product. Some texts discuss it with no explicit notation. Others use the subscripted "times" symbol (×_φ) as above to modify the direct product by inclusion of a homomorphism, writing the normal group on the left. Other notation reshapes the times symbol—for example: $\ltimes$ or $\rtimes$ , with or without subscripts. One way of thinking about the $\rtimes$ symbol is as a combination of the symbol for normal subgroup and the symbol for the product.

Unicode [1] lists four variants:

	value	MathML	Unicode description
⋉	U022C9	ltimes	LEFT NORMAL FACTOR SEMIDIRECT PRODUCT
⋊	U022CA	rtimes	RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT
⋋	U022CB	lthree	LEFT SEMIDIRECT PRODUCT
⋌	U022CC	rthree	RIGHT SEMIDIRECT PRODUCT

Although the Unicode description of the rtimes symbol says "right normal factor", a number of authors use it with a left normal factor. In Computing, Unicode is an Industry standard allowing Computers to consistently represent and manipulate text expressed in most of the world's Therefore the usual caution for mathematical notation applies: When reading, be careful to notice the conventions adopted by the author, and when writing, explain notation choices for the reader. The choice of symbol may vary, but putting the normal factor on the left seems fairly consistent.

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