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Homomorphisms are the maps between algebraic objects. There are two main types: group homomorphisms and ring homomorphisms. (Other examples include vector spacehomomorphisms, which are generally called linear maps, as well as homomorphisms of modules and homomorphisms of algebras.)

Generally speaking, a homomorphism between two algebraic objects A, B is a function f : A → B which preserves the algebraic structure on A and B That is, if elements in A satisfy some algebraic equation involving addition or multiplication, their images in B satisfy the same algebraic equation. The details of the definitions of homomorphisms in various contexts depend on the algebraic structures of A and B.

EXAMPLE

If the operations on A and B are both addition, then the homomorphism condition is  Homomorphism, Group Theory | Algebra - Mathematics  If A and B are both rings, with addition and multiplication, there is also a multiplicative condition:   Homomorphism, Group Theory | Algebra - Mathematics

A bijective homomorphism is called an isomorphism. An isomorphism between two algebraic objects  and  identifies them with each other; they are, in an algebraic sense, the same object (possibly written in two different ways). The most common use of homomorphisms in abstract algebra is via the three so-called isomorphism theorems, which allow for the identification of certain quotient objects with certain other subobjects (subgroups, subrings, etc.)

The study of the interplay between algebraic objects is fundamental in the study of algebra. The existence and properties of homomorphisms from one algebraic object to another give a rich depth of information about the objects and their relationship. Many important concepts in abstract algebra, such as

  • the integers modulo n

  • a prime ideal in a ring

  • the sign of a permutation,

can be naturally considered as (respectively) the image of a homomorphism, the kernel of a homomorphism, or the homomorphism itself.

Definitions and Examples

Let A and B be groups, with operations given by  ºA and ºB respectively. A group homomorphism f : A → B is a function f such that  Homomorphism, Group Theory | Algebra - Mathematics  for all x,y ∈ A.

 

DEFINATION

Let R and S be rings, with operations + and . (this is a slight abuse of notation, but the formulas below are more unwieldy with subscripts on the operations). A ring homomorphism f : R → S is a function f such that

Homomorphism, Group Theory | Algebra - Mathematics

(In this wiki, "ring" means "ring with unity"; a homomorphism of rings is defined in the same way, but without the third condition.)

In both cases, a homomorphism is called an isomorphism if it is bijective.

EXAMPLE

Show that if f : R → S is a ring homomorphism, f(0R) = os.

Note that  Homomorphism, Group Theory | Algebra - Mathematics  by the homomorphism property. Since f(0R) has an additive inverse in S, we can add it to both sides of this equation to get 0S =  f(0R).

 

EXAMPLE

1. For any groups G and H, there is a trivial homomorphis Homomorphism, Group Theory | Algebra - Mathematics2. Let  be a positive integer. The functionHomomorphism, Group Theory | Algebra - Mathematics is a ring homomorphism (and as such, it is a homomorphism of additive groups).3. Define Homomorphism, Group Theory | Algebra - Mathematics is complex conjugation. Then c is a homomorphism from  C to itself. It is clearly a bijection, so it is in fact an isomorphism from C to itself.4. Let R be a subring of S, and pick Homomorphism, Group Theory | Algebra - Mathematics Then there is an evaluation homomorphism Homomorphism, Group Theory | Algebra - Mathematics is the ring of polynomials with coefficients in R.It is given by Homomorphism, Group Theory | Algebra - Mathematics5. The map  Homomorphism, Group Theory | Algebra - Mathematics is a group homomorphism. Note that R is an additive group and R* the set of nonzero real numbers, is a multiplicative group. The verification that f is a group homomorphism is precisely the law of exponents: Homomorphism, Group Theory | Algebra - Mathematics6. Let Sn be the symmetric group on n letters. There is a unique nontrivial group homomorphism Homomorphism, Group Theory | Algebra - Mathematics the latter being a group under multiplication. The value Homomorphism, Group Theory | Algebra - Mathematics is called the sign of σ, and is important in many applications, including one definition of the determinant of a matrix.

Kernel and Image

Any homomorphism f : A → B has two objects associated to it: the kernel, which is a subset of A, and the image, which is a subset of B.

DEFINATION

Homomorphism, Group Theory | Algebra - Mathematics  be a group homomorphism. The kernel of f, ker (f), is the subset of G consisting of elements G such that Homomorphism, Group Theory | Algebra - Mathematics is the group identity element).

Homomorphism, Group Theory | Algebra - Mathematics be a ring homomorphism. The kernel of  Homomorphism, Group Theory | Algebra - Mathematics is the subset of  R consisting of elements R such that Homomorphism, Group Theory | Algebra - Mathematics

For further exploration of the kernel in the setting of vector spaces, see the wiki.

The kernel of a homomorphism is an important object, in both group and ring theory. The following theorem identifies what kind of object it is:

EXAMPLE

Continuing the six examples above:

1. If Homomorphism, Group Theory | Algebra - Mathematics  is the trivial homomorphism, then ker Homomorphism, Group Theory | Algebra - Mathematics the trivial subgroup of  H2. The kernel of reduction mod n is the ideal Homomorphism, Group Theory | Algebra - Mathematics consisting of multiples of n. The image is all of Zn; reduction mod n is surjective.
3. The kernel of complex conjugation is {0}, the trivial ideal of C (Note that 0 is always in the kernel of a ring homomorphism, by the above example.) The image is all of C.
4. The kernel of evaluation at α is the set of polynomials with coefficients in R which vanish at α. This ideal is not always easy to determine, depending on the nature of R and S. To take a common example, suppose  Homomorphism, Group Theory | Algebra - Mathematics Which polynomials with rational coefficients vanish on Homomorphism, Group Theory | Algebra - Mathematics (See the algebraic number theory wiki for an answer.)The image of evaluation at α is a ring called R[α], which is a subring of S consisting of polynomials in α with coefficients in R.5. The kernel of exponentiation is the set of elements which map to the identity element of R*, which is 1 So the kernel is {0}. And the image of exponentiation is the subgroup Homomorphism, Group Theory | Algebra - Mathematics of positive real numbers.6. The kernel of the sign homomorphism is known as the alternating group AnIt is an important subgroup of Sn which furnishes examples of simple groups for  Homomorphism, Group Theory | Algebra - Mathematics The image of the sign homomorphism is Homomorphism, Group Theory | Algebra - Mathematics  since the sign is a nontrivial map, so it takes on both Homomorphism, Group Theory | Algebra - Mathematics for certain permutations.

Properties of Homomorphisms

  • Composition: The composition of homomorphisms is a homomorphism. That is, if Homomorphism, Group Theory | Algebra - Mathematics  are homomorphisms, then Homomorphism, Group Theory | Algebra - Mathematics is a homomorphism as well.
  • Isomorphisms: If f is an isomorphism, which is a bijective homomorphism, thenHomomorphism, Group Theory | Algebra - Mathematics is also a homomorphism. (Compare with homeomorphism, a similar concept in topology, which is a continuous function with a continuous inverse; a bijective continuous function does not necessarily have a continuous inverse.) 
  • Injectivity and the kernel: A group homomorphism f is injective if and only if its kernel  ker(f) equals {1}, where denotes the identity element of the domain. A ring homomorphism is injective if and only if its kernel equals {0} where 0 denotes the additive identity of the domain.
  • Field homomorphisms: If R is a field and S is not the zero ring, then any homomorphism  Homomorphism, Group Theory | Algebra - Mathematics is injective. (Proof: the kernel is an ideal, and the only ideals in a  field are the entire field and the zero ideal. Since  Homomorphism, Group Theory | Algebra - Mathematics it must be the latter.)
The document Homomorphism, Group Theory | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Homomorphism, Group Theory - Algebra - Mathematics

1. What is a homomorphism in group theory?
Ans. A homomorphism is a mathematical function that preserves the algebraic structure between two groups. In group theory, a homomorphism maps elements from one group to another in such a way that the operation on the first group is preserved in the second group. It is an important concept used to analyze the relationship between different groups.
2. How does a homomorphism relate to the concept of isomorphism?
Ans. A homomorphism is a more general concept compared to isomorphism in group theory. While an isomorphism is a bijective homomorphism, meaning it is both injective and surjective, a homomorphism does not necessarily have to be bijective. In other words, an isomorphism is a special case of a homomorphism where the function is both one-to-one and onto.
3. What are the applications of homomorphisms in mathematics?
Ans. Homomorphisms have various applications in mathematics. Some of the common applications include: - Homomorphisms are used to prove properties and theorems in group theory. - They are used to study the structure and symmetry of objects in abstract algebra. - Homomorphisms play a crucial role in cryptography and coding theory. - They are used in the analysis of topological spaces and algebraic structures.
4. Can you provide an example of a homomorphism in group theory?
Ans. Sure! Consider two groups, G and H, where G is the group of integers under addition and H is the group of positive integers under multiplication. The function f: G -> H defined by f(x) = 2^x is a homomorphism. It preserves the group operation, as f(a + b) = f(a) * f(b) for all a, b in G.
5. How is the concept of homomorphism useful in solving mathematical problems?
Ans. Homomorphisms provide a way to analyze the structure and properties of groups by establishing relationships between them. They allow us to identify common patterns and similarities between different groups, which can lead to the discovery of important theorems and techniques. By studying the homomorphisms, mathematicians can gain insights into the behavior of groups and use them to solve complex mathematical problems in various areas of mathematics.
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