Homomorphisms are the maps between algebraic objects. There are two main types: group homomorphisms and ring homomorphisms. (Other examples include vector space homomorphisms, which are generally called linear maps, as well as homomorphisms of modules and homomorphisms of algebras.)
Generally speaking, a homomorphism between two algebraic objects is a function which preserves the algebraic structure on and That is, if elements in satisfy some algebraic equation involving addition or multiplication, their images in satisfy the same algebraic equation. The details of the definitions of homomorphisms in various contexts depend on the algebraic structures of and
If the operations on and are both addition, then the homomorphism condition is If and are both rings, with addition and multiplication, there is also a multiplicative condition:
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
can be naturally considered as (respectively) the image of a homomorphism, the kernel of a homomorphism, or the homomorphism itself.
Let and be groups, with operations given by and respectively. A group homomorphism is a function such that for all
Let and 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 is a function such that
(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.
Show that if is a ring homomorphism,
Note that by the homomorphism property. Since has an additive inverse in we can add it to both sides of this equation to get
For any groups and there is a trivial homomorphism given by for all
Let be a positive integer. The function defined by is a ring homomorphism (and as such, it is a homomorphism of additive groups).
Define by i.e. is complex conjugation. Then is a homomorphism from to itself. It is clearly a bijection, so it is in fact an isomorphism from to itself.
Let be a subring of , and pick Then there is an evaluation homomorphism where is the ring of polynomials with coefficients in It is given by
The map defined by is a group homomorphism. Note that is an additive group and the set of nonzero real numbers, is a multiplicative group. The verification that is a group homomorphism is precisely the law of exponents:
Let be the symmetric group on letters. There is a unique nontrivial group homomorphism the latter being a group under multiplication. The value for is called the sign of and is important in many applications, including one definition of the determinant of a matrix.
Which of the following function(s) define group homomorphisms?
I. defined by
II. defined by
III. defined by
is the group of invertible matrices with real entries, with the operation being matrix multiplication.
is the additive group of the integers.
is the additive group of the integers modulo 4, whose elements are
Any homomorphism has two objects associated to it: the kernel, which is a subset of and the image, which is a subset of
Let be a group homomorphism. The kernel of is the subset of consisting of elements such that (where is the group identity element).
Let be a ring homomorphism. The kernel of is the subset of consisting of elements such that
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:
Let be a group homomorphism. Then is a normal subgroup of and is a subgroup of
Let be a ring homomorphism. Then is an ideal of and is a subring of
Continuing the six examples above:
If is the trivial homomorphism, then and the trivial subgroup of
The kernel of reduction mod is the ideal consisting of multiples of The image is all of ; reduction mod is surjective.
The kernel of complex conjugation is the trivial ideal of (Note that is always in the kernel of a ring homomorphism, by the above example.) The image is all of
The kernel of evaluation at is the set of polynomials with coefficients in which vanish at This ideal is not always easy to determine, depending on the nature of and To take a common example, suppose Which polynomials with rational coefficients vanish on ? (See the algebraic number theory wiki for an answer.)
The image of evaluation at is a ring called which is a subring of consisting of polynomials in with coefficients in
The kernel of exponentiation is the set of elements which map to the identity element of which is So the kernel is And the image of exponentiation is the subgroup of positive real numbers.
The kernel of the sign homomorphism is known as the alternating group It is an important subgroup of which furnishes examples of simple groups for The image of the sign homomorphism is since the sign is a nontrivial map, so it takes on both and for certain permutations.
Composition: The composition of homomorphisms is a homomorphism. That is, if and are homomorphisms, then is a homomorphism as well.
Isomorphisms: If is an isomorphism, which is a bijective homomorphism, then 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 is injective if and only if its kernel equals where denotes the identity element of the domain. A ring homomorphism is injective if and only if its kernel equals where denotes the additive identity of the domain.
Field homomorphisms: If is a field and is not the zero ring, then any homomorphism is injective. (Proof: the kernel is an ideal, and the only ideals in a field are the entire field and the zero ideal. Since it must be the latter.)