| Basic notions in group theory | | category of groups | | 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. A semidirect product is a generalization of a direct product. A semidirect product is a cartesian product as a set, but with a particular multiplication operation. In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
The word kernel has several meanings in mathematics, some related to each other and some not. ...
In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...
In mathematics, one can often define a direct product of objects already known, giving a new one. ...
In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...
In mathematics, a finite group is a group which has finitely many elements. ...
In mathematics, a discrete group is a group G equipped with the discrete topology. ...
In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...
In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ...
An additive group is a group, and any group can be written as an additive group, so the adjective additive does not describe a class of groups, but rather the notation used to write the group operation. ...
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
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 element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of...
In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ...
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 realized by Galois theory. ...
For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
In mathematics, one can often define a direct product of objects already known, giving a new one. ...
In mathematics, the Cartesian product is a direct product of sets. ...
Some equivalent definitions Let G be a group, N a normal subgroup of G (i.e., N ◁ G) and H a subgroup of G. The following statements are equivalent: In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
- 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. For other uses, see identity (disambiguation). ...
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 a way that respects the given group operations. ...
In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
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 fails to be injective. ...
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, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...
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. This remark leads to an extension problem, of describing the possibilities. In mathematics, one can often define a direct product of objects already 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 a way that respects the given group operations. ...
In group theory, if the factor group G/K is isomorphic to H, one says that G is an extension of H by K. To consider some examples, if G = H × K, then G is an extension of both H and K. More generally, if G is a semidirect product...
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. The map φ : H → Aut(N) defined by φ(h) = φh, where φh(n) = hnh-1 for all h in H and n in N, is a group homomorphism. Together N, H and φ determine G up to isomorphism, as we show now. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
Look up Up to on Wiktionary, the free dictionary 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  (or simply N ×φ H) is called the semidirect product of N and H with respect to φ, defined as follows. As a set, is defined as the cartesian product N × H. Multiplication of elements in the cartesian product is determined by the homomorphism φ, with the operation * defined by Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...
In mathematics, the Cartesian product is a direct product of sets. ...
 for all n1, n2 in N and h1, h2 in H. This is a group in which the identity element is (eN, eH) and the inverse of the element (n, h) is (φh–1(n–1), h–1). Pairs (n,eH) form a normal subgroup isomorphic to N, while pairs (eN, 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) = φh, where
 for all n in N and h in H. Then G is isomorphic to the semidirect product  ; the isomorphism sends the product nh to the tuple (n,h). In G, we have the multiplication rule  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 In mathematics, and more specifically in homological algebra, the splitting lemma states that the following statements regarding the below short exact sequence in any abelian category are equivalent: there exists a map t: B → A such that tq is the identity on A, there exists a map u: C → B...
In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...
 and a group homomorphism γ : H → G such that  , the identity map on H. In this case, φ : H → Aut(N) is given by φ(h) = φh, where An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...
- φh(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 is the direct product  .
Examples The dihedral group Dn with 2n elements is isomorphic to a semidirect product of the cyclic groups Cn and C2. Here, the non-identity element of C2 acts on Cn by inverting elements; this is an automorphism since Cn is abelian. The presentation for this group is: This article may be confusing for some readers, and should be edited to enhance clarity. ...
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 element g (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of...
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ...
In mathematics, one method of defining a group is by a presentation. ...
 . More generally, a semidirect product of any two cyclic groups  with generator  and  with generator  is given by a single relation  with  and  coprime, i.e. the presentation: In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
 . If  and  are coprime,  is a generator of and  , hence the presentation:  gives a group isomorphic to the previous one.
The fundamental group of the Klein bottle can be presented in the form In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
The Klein bottle immersed in three-dimensional space. ...
 and is therefore a semidirect product of the group of integers,  , with itself.
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). 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). Every orthogonal matrix acts as an automorphism on R2 by matrix multiplication. In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...
In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces. ...
In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
In a group, the conjugate by g of h is ghg-1. ...
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. 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"). In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...
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 objects already 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) = idN 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. There is also a construction in ring theory, the crossed product of rings. This is seen naturally as soon as one constructs a group ring for a semidirect product of groups. There is also the semidirect sum of Lie algebras. 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. 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. noncommutative geometry). 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 from two subgroups. ...
In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...
In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...
In mathematics, a symmetry group describes all symmetries of objects. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
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 Neumann algebra acted on by a group. ...
Alain Connes (born April 1, 1947) is a French mathematician, currently Professor at the College de France (Paris, France), IHES (Bures-sur-Yvette, France) and Vanderbilt University (Nashville, Tennessee). ...
In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ...
There are also far-reaching generalisations in category theory. They show how to construct fibred categories from indexed categories. This is an abstract form of the outer semidirect product construction. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, specifically in category theory, a fibred category or fibered category, is a functor such that every morphism in has a unique cartesian lift. ...
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:  or  , with or without subscripts. One way of thinking about the symbol is as a combination of the symbol for normal subgroup and the symbol for the product.
Unicode [1] lists four variants: The Unicode Standard, Version 5. ...
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| | | 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. 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.
In LaTeX, the commands rtimes and ltimes produce the corresponding characters. This article is about the typesetting system. ...
See also In mathematics, one can often define a direct product of objects already known, giving a new one. ...
In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. ...
// as a semi-direct product In group theory, for a group , the holomorph of denoted can be described in two ways. ...
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