1 Definition and basic idea
A homomorphism is a map between algebraic structures that preserves the operations defining those structures. The term is used across many branches of algebra, including group theory, ring theory, module theory, and linear algebra. In each setting, the core idea is the same: applying the map after combining elements gives the same result as combining their images in the corresponding way.
Homomorphisms are useful because they let mathematicians study one object through another. They often reveal hidden similarities between different systems and make it possible to transfer results from a simpler setting to a more complicated one.
1.1 Structure-preserving maps
A structure-preserving map respects the basic rules of an algebraic system. For a group, this means preserving the group operation. For a ring, it means preserving addition and multiplication. For a vector space, it means preserving addition and scalar multiplication. The exact requirement depends on the structure being considered, but the guiding principle is always compatibility with the relevant operations.
Such maps can collapse distinct elements into the same image, or they can embed one structure into another without losing information. The extent to which they preserve information is measured by the kernel and image.
1.2 Homomorphism notation
Homomorphisms are often written using letters such as \(f\), \(g\), or \(\varphi\). If \(f\) is a homomorphism from a structure \(A\) to a structure \(B\), one may write \(f: A \to B\). When the structure is important, the notation may include the type of homomorphism, such as group homomorphism or ring homomorphism.
The image of an element \(a\) under \(f\) is usually written \(f(a)\). For two elements \(a\) and \(b\), the preservation property is expressed in the form \(f(a \ast b) = f(a) \ast f(b)\), where \(\ast\) denotes the relevant operation.
1.3 Examples of homomorphisms
Homomorphisms appear in many familiar algebraic contexts. Some of the most common examples come from groups, rings, and vector spaces. These examples illustrate how the same abstract idea adapts to different kinds of structure.
1.3.1 Group homomorphisms
A group homomorphism preserves the group operation. For example, the map from the integers under addition to the integers modulo \(n\) given by reduction modulo \(n\) is a group homomorphism. It sends sums to sums taken modulo \(n\).
1.3.2 Ring homomorphisms
A ring homomorphism preserves both addition and multiplication. The natural projection from the integers to the integers modulo \(n\) is a standard example. It carries sums and products to their corresponding residue classes.
1.3.3 Vector space linear maps
A linear map between vector spaces preserves addition and scalar multiplication. Such maps are the linear algebra counterpart of homomorphisms. Matrix multiplication provides many examples of linear maps, since a matrix defines a transformation that respects the vector space operations.
2 Homomorphisms by algebraic structure
Different algebraic systems require different preservation laws. Although the terminology changes from one field to another, each notion of homomorphism is a specialized version of the same underlying concept.
2.1 Group homomorphisms
A group homomorphism is a map between groups that respects the group operation. Since groups may be written additively or multiplicatively, the formula is adjusted to the chosen notation, but the meaning remains the same.
2.1.1 Compatibility with the group operation
If \(f\) is a group homomorphism, then \(f(ab) = f(a)f(b)\) in multiplicative notation, or \(f(a+b) = f(a) + f(b)\) in additive notation. This compatibility ensures that the group law is carried faithfully through the map.
2.1.2 Identity and inverse elements
A group homomorphism sends the identity element of the source group to the identity element of the target group. It also sends inverses to inverses. These facts follow directly from operation preservation and are fundamental in studying group structure.
2.2 Ring homomorphisms
A ring homomorphism preserves the additive and multiplicative structure of rings. Because rings combine two operations, homomorphisms between them must respect both laws simultaneously.
2.2.1 Addition and multiplication preservation
If \(f\) is a ring homomorphism, then \(f(a+b)=f(a)+f(b)\) and \(f(ab)=f(a)f(b)\). These conditions ensure that sums, products, and derived expressions behave consistently under the map.
2.2.2 Unital and non-unital homomorphisms
Some definitions of ring homomorphism require the map to send the multiplicative identity of the source to the multiplicative identity of the target. These are called unital homomorphisms. Other contexts allow homomorphisms that do not preserve identity elements. The convention depends on the algebraic setting and the author’s choice.
2.3 Module homomorphisms
A module homomorphism is a map between modules that respects both addition and scalar multiplication by the underlying ring. It generalizes linear maps from vector spaces, where the scalars come from a field rather than a ring.
2.3.1 Additivity and scalar compatibility
If \(f\) is a module homomorphism, then \(f(m_1+m_2)=f(m_1)+f(m_2)\) and \(f(rm)=r f(m)\) for scalars \(r\) and module elements \(m\). These two rules capture the module structure and make the map compatible with scalar action.
2.4 Lie algebra homomorphisms
Lie algebra homomorphisms preserve the Lie algebra operations, especially the Lie bracket. They are central in the study of symmetries and infinitesimal transformations.
2.4.1 Preservation of the Lie bracket
If \(f\) is a Lie algebra homomorphism, then \(f([x,y]) = [f(x),f(y)]\). This requirement ensures that the commutator-like structure encoded by the bracket is maintained under the map.
2.5 Algebra homomorphisms
An algebra homomorphism is a map between algebras that preserves the operations of addition, multiplication, and scalar multiplication. It combines the conditions for ring homomorphisms and module homomorphisms.
2.5.1 Compatibility with both algebraic and scalar operations
For an algebra homomorphism, the map must be linear over the base ring or field and must also preserve products. This dual compatibility makes algebra homomorphisms especially important in studying polynomial algebras, matrix algebras, and representation-theoretic constructions.
3 Fundamental concepts
Several concepts are used repeatedly when analyzing homomorphisms. Among the most important are the kernel, the image, and the special case of an isomorphism.
3.1 Kernel
The kernel measures which elements are collapsed to the neutral element by a homomorphism. It is a central tool for understanding how much information a map loses.
3.1.1 Definition of the kernel
The kernel of a homomorphism consists of all elements in the source that map to the identity element of the target, or to zero in additive settings. It is often denoted by \(\ker(f)\).
3.1.2 Relation to injectivity
A homomorphism is injective if and only if its kernel is trivial, meaning it contains only the identity element or only zero. This relationship makes the kernel a convenient criterion for detecting whether a homomorphism preserves distinct elements.
3.2 Image
The image is the set of all outputs produced by a homomorphism. It describes the portion of the target structure that is actually reached by the map.
3.2.1 Definition of the image
The image of \(f\) is the set \(\{f(a) : a \in A\}\). It is denoted by \(\operatorname{im}(f)\) or \(f(A)\). By construction, every element of the image comes from some element of the source.
3.2.2 Substructure properties of the image
The image of a homomorphism usually forms a substructure of the target, such as a subgroup, subring, submodule, or subspace. This is one reason homomorphisms are so effective: they naturally produce smaller structures inside larger ones.
3.3 Isomorphisms
An isomorphism is a homomorphism that preserves all structure without losing any information. It identifies two algebraic objects as essentially the same from the standpoint of the relevant operations.
3.3.1 Bijective homomorphisms
A bijective homomorphism is an isomorphism in the usual algebraic sense. Since it is both injective and surjective, it matches every element of the source with exactly one element of the target.
3.3.2 Inverse homomorphisms
The inverse of an isomorphism is again a homomorphism. This property shows that isomorphic structures can be translated back and forth without altering the algebraic content.
4 Properties and theorems
Homomorphisms satisfy a range of structural results that make them indispensable in algebra. These results explain how substructures, products, and quotient constructions behave under mapping.
4.1 Preservation of structure
Because homomorphisms respect operations, they often preserve more elaborate constructions built from those operations. This includes relationships involving subobjects, products, and identities.
4.1.1 Subobjects and products
Homomorphisms interact naturally with subgroups, subrings, submodules, and related objects. They also behave well with product structures, where maps can be defined coordinatewise. Such compatibility helps organize algebraic systems into families connected by simple maps.
4.1.2 Identities and compositions
The identity map on a structure is always a homomorphism. The composition of two homomorphisms is again a homomorphism. These facts are basic but fundamental, since they allow homomorphisms to be chained together in a controlled way.
4.2 First isomorphism theorem
The first isomorphism theorem connects a homomorphism with its kernel and image. It is one of the most important results in algebra because it explains how every homomorphism factors through a quotient.
4.2.1 Quotients by kernels
The theorem states, in a broad sense, that the source modulo the kernel is isomorphic to the image. This identifies the kernel as the precise obstruction to injectivity and shows that collapsing the kernel captures exactly the information retained by the map.
4.2.2 Canonical factorization
Every homomorphism can be decomposed into a projection onto a quotient followed by an injective map into the target. This factorization is canonical, meaning it arises naturally from the given homomorphism rather than from an arbitrary choice.
4.3 Second and third isomorphism theorems
The second and third isomorphism theorems describe how quotient structures interact with substructures. They provide a framework for comparing nested algebraic objects.
4.3.1 Relations among quotient structures
These theorems identify relationships between quotients formed from substructures and larger ambient structures. They often show that a quotient taken in one order is naturally isomorphic to a quotient taken in another order.
4.3.2 Correspondence results
The theorems also support correspondence between substructures in a quotient and substructures in the original object. This correspondence is especially useful in group theory and module theory, where it clarifies how intermediate structures fit together.
5 Categorical viewpoint
In category theory, homomorphisms are understood as morphisms, the arrows that connect objects in a category. This perspective emphasizes the role of maps and composition rather than the internal details of any one algebraic system.
5.1 Morphisms in category theory
A morphism is an abstract map satisfying the structural rules of a category. In algebraic categories, morphisms are usually precisely the homomorphisms of the relevant type. This unifies many branches of algebra under a single language.
5.2 Endomorphisms and automorphisms
Special kinds of homomorphisms arise when the source and target are the same or when the map is invertible. These cases are important because they describe internal symmetries and self-transformations.
5.2.1 Self-maps
An endomorphism is a homomorphism from an algebraic object to itself. Endomorphisms form a rich collection of transformations that can reveal internal patterns, fixed points, and invariant substructures.
5.2.2 Structure-preserving bijections
An automorphism is a bijective endomorphism. It gives a symmetry of the structure, sending it onto itself while preserving all operations. Automorphisms are often studied as a group under composition.
5.3 Composition of homomorphisms
Composition is the process of applying one homomorphism after another. It is central to the categorical and algebraic use of maps.
5.3.1 Associativity of composition
Composition of homomorphisms is associative: if three homomorphisms can be composed, the order of grouping does not affect the result. This property is essential for building longer chains of maps.
5.3.2 Identity morphisms
Every object has an identity morphism that acts as a neutral element for composition. When composed with any homomorphism with matching domain or codomain, it leaves that homomorphism unchanged.
6 Special cases and examples
Certain homomorphisms appear so often that they are treated as special examples. They include constant maps, inclusion maps, projections, and evaluation maps.
6.1 Constant and zero homomorphisms
A constant homomorphism sends every element to a fixed value. In many algebraic contexts, the only constant homomorphism is the one sending everything to the neutral element, often called the zero homomorphism. This map is always a homomorphism when the algebraic rules permit it.
6.2 Inclusion and projection maps
An inclusion map treats a substructure as part of a larger one. It is typically a homomorphism that preserves the operations exactly as they appear in the ambient structure. A projection map selects one component from a product or direct sum and is also a standard example of a homomorphism.
6.3 Evaluation homomorphisms
An evaluation homomorphism sends an algebraic expression to its value at a specified input. For example, evaluating a polynomial at a chosen element defines a homomorphism in many common settings. These maps connect symbolic algebra with concrete numerical or structural results.
6.4 Universal properties
Universal properties characterize objects by the maps they receive or send. Homomorphisms are often the language in which these properties are stated. This makes them essential in defining free objects, quotient objects, products, and other constructions in an abstract but precise way.
7 Applications
Homomorphisms are used throughout algebra because they simplify complex structures and expose their essential features. They are a primary tool for construction, classification, and proof.
7.1 Constructing quotient structures
Quotient structures are built by identifying elements that a homomorphism sends to the same value. The kernel determines the equivalence relation used in the quotient. This process produces a new structure in which certain distinctions have been intentionally removed.
7.2 Comparing algebraic systems
Homomorphisms provide a way to compare one algebraic system with another. If a map is injective, it embeds one system inside another. If it is surjective, it shows that one system can be realized as a homomorphic image of another. If it is an isomorphism, it shows that the two systems are structurally identical.
7.3 Solving equations in algebra
Homomorphisms help simplify equations by passing them into more manageable structures. Because they preserve operations, solutions in the source often correspond to solutions in the image, subject to the kernel. This is especially useful in modular arithmetic, polynomial algebra, and group-theoretic computations.
7.4 Representation and module theory
In representation theory, homomorphisms describe how abstract algebraic objects act on vector spaces or modules. Module homomorphisms, in particular, govern the maps between representations and encode the ways structures can be transformed while respecting scalar action. These ideas provide a bridge between abstract algebra and linear methods.