Isomorphism theorem - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, the isomorphism theorems are three theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s that describe the relationship between quotient

Quotient

In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

s, homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

s, and subobject

Subobject

In category theory, a branch of mathematics, a subobject is, roughly speaking, an object which sits inside another object in the same category. The notion is a generalization of the older concepts of subset from set theory and subgroup from group theory...

s. Versions of the theorems exist for groups

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, rings

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

, vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s, modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

, Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

s, and various other algebraic structure

Algebraic structure

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

s. In universal algebra

Universal algebra

Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....

, the isomorphism theorems can be generalized to the context of algebras and congruences.

History

The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether

Emmy Noether

Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...

in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern which was published in 1927 in Mathematische Annalen

Mathematische Annalen

Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...

. Less general versions of these theorems can be found in work of Richard Dedekind

Richard Dedekind

Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

and previous papers by Noether.

Three years later, B.L. van der Waerden

Bartel Leendert van der Waerden

Bartel Leendert van der Waerden was a Dutch mathematician and historian of mathematics....

published his influential Algebra, the first abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

textbook that took the now-traditional groups

Group (mathematics)

-rings

Ring (mathematics)

-fields

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

approach to the subject. Van der Waerden credited lectures by Noether on group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

and Emil Artin

Emil Artin

Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...

on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke

Wilhelm Blaschke

Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....

, Otto Schreier

Otto Schreier

Otto Schreier was an Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. He studied mathematics at the University of Vienna and obtained his doctorate in 1923, under the supervision of Philipp Furtwängler...

, and van der Waerden himself on ideals

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

as the main references. The three isomorphism theorems, called homomorphism theorem, and two laws of isomorphism when applied to groups, appear explicitly.

Groups

We first state the three isomorphism theorems in the context of groups

Group (mathematics)

. Note that some sources switch the numbering of the second and third theorems. Sometimes, the lattice theorem

Lattice theorem

In mathematics, the lattice theorem, sometimes referred to as the fourth isomorphism theorem or the correspondence theorem, states that if N is a normal subgroup of a group G, then there exists a bijection from the set of all subgroups A of G such that A contains N, onto the set of all subgroups...

is referred to as the fourth isomorphism theorem or the correspondence theorem.

First isomorphism theorem

Let G and H be groups, and let φ: G → H be a homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

. Then:

The kernel
Kernel (algebra)
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. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

of φ is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

of G,
The image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

of φ is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...

of H, and
The image of φ is isomorphic
Group isomorphism
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. If there exists an isomorphism between two groups, then the groups are called isomorphic...

to the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

G / ker(φ).

In particular, if φ is surjective

Surjective function

In mathematics, a function f from a set X to a set Y is surjective , or a surjection, if every element y in Y has a corresponding element x in X so that f = y...

then H is isomorphic to G / ker(φ).

Second isomorphism theorem

Let G be a group. Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

The product SN is a subgroup of G,
The intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

S ∩ N is a normal subgroup of S, and
The quotient groups (SN) / N and S / (S ∩ N) are isomorphic.

Technically, it is not necessary for N to be a normal subgroup, as long as S is a subgroup of the normalizer of N. In this case, the intersection S ∩ N is not a normal subgroup of G, but it is still a normal subgroup of S.

Third isomorphism theorem

Let G be a group. Let N and K be normal subgroups of G, with

K ⊆ N ⊆ G.

Then

The quotient N / K is a normal subgroup of the quotient G / K, and
The quotient group (G / K) / (N / K) is isomorphic to G / N.

Discussion

**First isomorphism theorem**

The first isomorphism theorem follows from the category theoretical

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

fact that the category of groups

Category of groups

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...

is (normal epi, mono)-factorizable; in other words, the normal epimorphisms

Normal morphism

In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism.A normal category is a category in which every monomorphism is normal...

and the monomorphism

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

s form a factorization system

Factorization system

In mathematics, it can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are a generalization of this situation in category theory.-Definition:...

for the category. This is captured in the commutative diagram

Commutative diagram

In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...

in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism f: G→H. The diagram shows that every morphism in the category of groups has a kernel

Kernel (category theory)

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...

in the category theoretical sense; the arbitrary morphism f factors into

, where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object

and a monomorphism

(kernels are always monomorphisms), which complete the short exact sequence

Exact sequence

An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...

running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphism

Zero morphism

In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

s from

to H and

.

If the sequence is right split (i. e., there is a morphism σ that maps

to a π-preimage of itself), then G is the semidirect product

Semidirect product

In mathematics, specifically 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...

of the normal subgroup

and the subgroup

. If it is left split (i. e., there exists some

such that

), then it must also be right split, and

is a direct product

Direct product

In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....

decomposition of G. In general, the existence of a right split does not imply the existence of a left split; but in an abelian category

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

(such as the abelian groups), left splits and right splits are equivalent by the splitting lemma

Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent....

, and a right split is sufficient to produce a direct sum decomposition

. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence

.

In the second isomorphism theorem, the product SN is the join of S and N in the lattice of subgroups

Lattice of subgroups

In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion....

of G, while the intersection S ∩ N is the meet

Meet (mathematics)

In mathematics, join and meet are dual binary operations on the elements of a partially ordered set. A join on a set is defined as the supremum with respect to a partial order on the set, provided a supremum exists...

.

The third isomorphism theorem is generalized by the nine lemma

Nine lemma

In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups. It states: ifis a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well...

to abelian categories and more general maps between objects. It is sometimes informally called the "freshman theorem", because "even a freshman could figure it out: just cancel out the Ks!"

Rings

The statements of the theorems for rings

Ring (mathematics)

are similar, with the notion of a normal subgroup replaced by the notion of an ideal

Ideal (ring theory)

First isomorphism theorem

Let R and S be rings, and let φ: R → S be a ring homomorphism

Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....

. Then:

The kernel
Kernel (algebra)
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. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

of φ is an ideal of R,
The image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

of φ is a subring
Subring
In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...

of S, and
The image of φ is isomorphic to the quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

R / ker(φ).

In particular, if φ is surjective

Surjective function

In mathematics, a function f from a set X to a set Y is surjective , or a surjection, if every element y in Y has a corresponding element x in X so that f = y...

then S is isomorphic to R / ker(φ).

Second isomorphism theorem

Let R be a ring. Let S be a subring of R, and let I be an ideal of R. Then:

The sum S + I = {s + i | s ∈ S, i ∈ I} is a subring of R,
The intersection S ∩ I is an ideal of S, and
The quotient rings (S + I) / I and S / (S ∩ I) are isomorphic.

Third isomorphism theorem

Let R be a ring. Let A and B be ideals of R, with

B ⊆ A ⊆ R.

Then

The set A / B is an ideal of the quotient R / B, and
The quotient ring (R / B) / (A / B) is isomorphic to R / A.

Modules

The statements of the isomorphism theorems for modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

are particularly simple, since it is possible to form a quotient module

Quotient module

In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic...

from any submodule. The isomorphism theorems for vector space

Vector space

s and abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s are special cases of these. For vector spaces, all of these theorems follow from the rank-nullity theorem

Rank-nullity theorem

In mathematics, the rank–nullity theorem of linear algebra, in its simplest form, states that the rank and the nullity of a matrix add up to the number of columns of the matrix. Specifically, if A is an m-by-n matrix over some field, thenThis applies to linear maps as well...

.

For all of the following theorems, the word “module” will mean “R-module”, where R is some fixed ring.

First isomorphism theorem

Let M and N be modules, and let φ: M → N be a homomorphism. Then:

The kernel
Kernel (algebra)
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. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

of φ is a submodule of M,
The image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

of φ is a submodule of N, and
The image of φ is isomorphic to the quotient module
Quotient module
In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic...

M / ker(φ).