Group homomorphism

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h : GH such that for all u and v in G it holds that

where the group operation on the left hand side of the equation is that of G and on the right hand side that of H.

From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H,

and it also maps inverses to inverses in the sense that

Hence one can say that h "is compatible with the group structure".

Older notations for the homomorphism h(x) may be xh or xh, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right.

In areas of mathematics where one considers groups endowed with additional structure, a homomorphism sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.

Group homomorphism ver.2
Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the kernel of h and aN is a coset of N.

Intuition

The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function h : GH is a group homomorphism if whenever

ab = c   we have   h(a) ⋅ h(b) = h(c).

In other words, the group H in some sense has a similar algebraic structure as G and the homomorphism h preserves that.

Types of group homomorphism

Monomorphism
A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness.
Epimorphism
A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain.
Isomorphism
A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups G and H are called isomorphic; they differ only in the notation of their elements and are identical for all practical purposes.
Endomorphism
A homomorphism, h: GG; the domain and codomain are the same. Also called an endomorphism of G.
Automorphism
An endomorphism that is bijective, and hence an isomorphism. The set of all automorphisms of a group G, with functional composition as operation, forms itself a group, the automorphism group of G. It is denoted by Aut(G). As an example, the automorphism group of (Z, +) contains only two elements, the identity transformation and multiplication with −1; it is isomorphic to Z/2Z.

Image and kernel

We define the kernel of h to be the set of elements in G which are mapped to the identity in H

and the image of h to be

The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem states that the image of a group homomorphism, h(G) is isomorphic to the quotient group G/ker h.

The kernel of h is a normal subgroup of G and the image of h is a subgroup of H:

If and only if ker(h) = {eG}, the homomorphism, h, is a group monomorphism; i.e., h is injective (one-to-one). Injection directly gives that there is a unique element in the kernel, and a unique element in the kernel gives injection:

Examples

  • Consider the cyclic group Z/3Z = {0, 1, 2} and the group of integers Z with addition. The map h : ZZ/3Z with h(u) = u mod 3 is a group homomorphism. It is surjective and its kernel consists of all integers which are divisible by 3.
  • Consider the group

    For any complex number u the function fu : GC defined by:

    is a group homomorphism.
  • Consider multiplicative group of positive real numbers (R+, ⋅) for any complex number u the function fu : R+C defined by:
    is a group homomorphism.
  • The exponential map yields a group homomorphism from the group of real numbers R with addition to the group of non-zero real numbers R* with multiplication. The kernel is {0} and the image consists of the positive real numbers.
  • The exponential map also yields a group homomorphism from the group of complex numbers C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and has the kernel {2πki : kZ}, as can be seen from Euler's formula. Fields like R and C that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields.

The category of groups

If h : GH and k : HK are group homomorphisms, then so is kh : GK. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.

Homomorphisms of abelian groups

If G and H are abelian (i.e., commutative) groups, then the set Hom(G, H) of all group homomorphisms from G to H is itself an abelian group: the sum h + k of two homomorphisms is defined by

(h + k)(u) = h(u) + k(u)    for all u in G.

The commutativity of H is needed to prove that h + k is again a group homomorphism.

The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if f is in Hom(K, G), h, k are elements of Hom(G, H), and g is in Hom(H, L), then

(h + k) ∘ f = (hf) + (kf)    and    g ∘ (h + k) = (gh) + (gk).

Since the composition is associative, this shows that the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. For example, the endomorphism ring of the abelian group consisting of the direct sum of m copies of Z/nZ is isomorphic to the ring of m-by-m matrices with entries in Z/nZ. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.

See also

References

  • Dummit, D. S.; Foote, R. (2004). Abstract Algebra (3rd ed.). Wiley. pp. 71–72. ISBN 978-0-471-43334-7.
  • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001

External links

Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion). However, the concept of formal correctness depends on time and on the context. Therefore, many notations in mathematics are qualified as abuse of notation in some context and are formally correct in other contexts; as many notations were introduced a long time before any formalization of the theory in which they are used, the qualification of abuse of notation is strongly time dependent. Moreover, many abuses of notation may be made formally correct by improving the theory. Abuse of notation should be contrasted with misuse of notation, which should be avoided.

A related concept is abuse of language or abuse of terminology, when not notation but a term is misused. Abuse of language is an almost synonymous expression that is usually used for non-notational abuses. For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V "a representation of G". A common abuse of language consists in identifying two mathematical objects that are different but canonically isomorphic. Examples include identifying a constant function and its value or identifying to the Euclidean space of dimension three equipped with a Cartesian coordinate system.

Additive map

In algebra, an additive map, Z-linear map or additive function is a function that preserves the addition operation:

for every pair of elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial.

More formally, an additive map is a Z-module homomorphism. Since an abelian group is a Z-module, it may be defined as a group homomorphism between abelian groups.

Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object, for example the product operation of a ring.

If f and g are additive maps, then the map f + g (defined pointwise) is additive.

A map V × WX that is additive each of two arguments separately is called a bi-additive map or a Z-bilinear map.

Affine

Affine may refer to:

Affine cipher, a special case of the more general substitution cipher

Affine combination, a certain kind of constrained linear combination

Affine connection, a connection on the tangent bundle of a differentiable manifold

Affine differential geometry, a geometry that studies differential invariants under the action of the special affine group

Affine geometry, a geometry characterized by parallel lines

Affine group, the group of all invertible affine transformations from any affine space over a field K into itself

Affine logic, a substructural logic whose proof theory rejects the structural rule of contraction

Affine representation, a continuous group homomorphism whose values are automorphisms of an affine space

Affine scheme, the spectrum of prime ideals of a commutative ring

Affine morphism, a morphism of schemes such that the pre-image of an open affine subscheme is affine

Affine space, an abstract structure that generalises the affine-geometric properties of Euclidean space

Affine transformation, a transformation that preserves the relation of parallelism between lines

Affine, a relative by marriage

Affine representation

An affine representation of a topological (Lie) group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A.

An example is the action of the Euclidean group E(n) upon the Euclidean space En.

Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space: in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general.

Category of abelian groups

In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.

Covering group

In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which H has index 2 in G; examples include the Spin groups, Pin groups, and metaplectic groups.

Roughly explained, saying that for example the metaplectic group Mp2n is a double cover of the symplectic group Sp2n means that there are always two elements in the metaplectic group representing one element in the symplectic group.

Direct limit

In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by . (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.)

Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits which are a special case of limits in category theory.

Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.

In any category, the composition of any two endomorphisms of X is again an endomorphism of X. It follows that the set of all endomorphisms of X forms a monoid, denoted End(X) (or EndC(X) to emphasize the category C).

Finitely generated object

In category theory, a finitely generated object is the quotient of a free object over a finite set, in the sense that it is the target of a regular epimorphism from a free object that is free on a finite set.For instance, one way of defining a finitely generated group is that it is the image of a group homomorphism from a finitely generated free group.

Functor

In mathematics, a functor is a map between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied.

The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap, who used the term in a linguistic context;

see function word.

Gauss sum

In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically

where the sum is over elements r of some finite commutative ring R, ψ is a group homomorphism of the additive group R+ into the unit circle, and χ is a group homomorphism of the unit group R× into the unit circle, extended to non-unit r, where it takes the value 0. Gauss sums are the analogues for finite fields of the Gamma function.[clarification needed]

Such sums are ubiquitous in number theory. They occur, for example, in the functional equations of Dirichlet L-functions, where for a Dirichlet character χ the equation relating L(s, χ) and L(1 − s, χ) (where χ is the complex conjugate of χ) involves a factor[clarification needed]

Homomorphism

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same".Homomorphisms of vector spaces are also called linear maps, and their study is the object of linear algebra.

The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.

A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.

Induced homomorphism

In mathematics, especially in the area of topology known as algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y.

More generally, in category theory, any functor by definition provides an induced morphism in the target category for each morphism in the source category. For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from the category of (e.g.) topological spaces to the category of (e.g.) groups or rings. This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism. A homomorphism induced from a map h is often denoted .

Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are inverse to each other up to homotopy induce homomorphisms that are inverse to each other. A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations between spaces and continuous maps, often very intricate, can be inferred from relations between the homomorphisms they induce. The latter may be simpler to analyze, since they involve algebraic structures which can be often easily described, compared, and calculated in.

Isogeny

In mathematics, an isogeny is a morphism of algebraic groups that is surjective and has a finite kernel.

If the groups are abelian varieties, then any morphism f : A → B of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.

The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind or nature". The term "isogeny" was introduced by Weil; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny.

Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are smooth. Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.

In rough terms, a Lie group is a continuous group, that is, one whose elements are described by several real parameters. As such, Lie groups provide a natural model for the concept of continuous symmetry, such as rotational symmetry in three dimensions. Lie groups are widely used in many parts of modern mathematics and physics. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism

from the real line (as an additive group) to some other topological group . That means that it is not in fact a group, strictly speaking; if is injective then , the image, will be a subgroup of that is isomorphic to as additive group.

One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations. According to Lie, an infinitesimal transformation is an infinitely small transformation of the one-parameter group that it generates. It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension.

Section conjecture

In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism , where is a complete smooth curve of genus at least 2 over a field that is finitely generated over , in terms of decomposition groups of rational points of . The conjecture was introduced by Alexander Grothendieck (1997) in a 1983 letter to Gerd Faltings.

Transfer (group theory)

In the mathematical field of group theory, the transfer defines, given a group G and a subgroup of finite index H, a group homomorphism from G to the abelianization of H. It can be used in conjunction with the Sylow theorems to obtain certain numerical results on the existence of finite simple groups.

The transfer was defined by Issai Schur (1902) and rediscovered by Emil Artin (1929).

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