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In mathematics, specifically
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, rings, vector spaces, modules, Lie algebras, and various other
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s. In universal algebra, 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 in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years later, B.L. van der Waerden published his influential '' Moderne Algebra'' the first
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
textbook that took the
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
- rings-
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
approach to the subject. Van der Waerden credited lectures by Noether on
group theory In 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 ...
and Emil Artin on algebra, as well as a seminar conducted by Artin, Wilhelm Blaschke, Otto Schreier, and van der Waerden himself on ideals 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 present the isomorphism theorems of the
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
.


Note on numbers and names

Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules. It is less common to include the Theorem D, usually known as the '' lattice theorem'' or the ''correspondence theorem'', as one of isomorphism theorems, but when included, it is the last one.


Statement of the theorems


Theorem A (groups)

Let ''G'' and ''H'' be groups, and let ''f'' : ''G'' → ''H'' be a homomorphism. Then: # The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of ''f'' is a normal subgroup of ''G'', # The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''f'' is a
subgroup In group theory, a branch of mathematics, 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 subgrou ...
of ''H'', and # The image of ''f'' is isomorphic to the quotient group ''G'' / ker(''f''). In particular, if ''f'' is surjective then ''H'' is isomorphic to ''G'' / ker(''f'').


Theorem B (groups)

Let G be a group. Let S be a subgroup of G, and let N be a normal subgroup of G. Then the following hold: # The
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
SN is a subgroup of G, # The intersection S \cap N is a normal subgroup of S, and # The quotient groups (SN)/N and S/(S\cap 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 G. In this case, the intersection S \cap N is not a normal subgroup of G, but it is still a normal subgroup of S. This theorem is sometimes called the ''isomorphism theorem'', ''diamond theorem'' or the ''parallelogram theorem''. An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the
complex projective line In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
starts with setting G = \operatorname_2(\mathbb), the group of invertible 2 × 2
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
matrices, S = \operatorname_2(\mathbb), the subgroup of determinant 1 matrices, and N the normal subgroup of scalar matrices \mathbb^\!I = \left\, we have S \cap N = \, where I is the identity matrix, and SN = \operatorname_2(\mathbb). Then the second isomorphism theorem states that: : \operatorname_2(\mathbb) := \operatorname_2 \left(\mathbb)/(\mathbb^\!I\right) \cong \operatorname_2(\mathbb)/\ =: \operatorname_2(\mathbb)


Theorem C (groups)

Let G be a group, and N a normal subgroup of G. Then # If K is a subgroup of G such that N \subseteq K \subseteq G, then G/N has a subgroup isomorphic to K/N. # Every subgroup of G/N is of the form K/N for some subgroup K of G such that N \subseteq K \subseteq G. # If K is a normal subgroup of G such that N \subseteq K \subseteq G, then G/N has a normal subgroup isomorphic to K/N. # Every normal subgroup of G/N is of the form K/N for some normal subgroup K of G such that N \subseteq K \subseteq G. # If K is a normal subgroup of G such that N \subseteq K \subseteq G, then the quotient group (G/N)/(K/N) is isomorphic to G/K.


Theorem D (groups)

The correspondence theorem (also known as the lattice theorem) is sometimes called the third or fourth isomorphism theorem. The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.


Discussion

The first isomorphism theorem can be expressed in category theoretical language by saying that the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories The ...
is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
s form a
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 A factoriza ...
for the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *C ...
. This is captured in the
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the ...
in the margin, which shows the
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
and
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s whose existence can be deduced from the morphism f : G \rightarrow H. The diagram shows that every morphism in the category of groups has a
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
in the category theoretical sense; the arbitrary morphism ''f'' factors into \iota \circ \pi, 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 \ker f and a monomorphism \kappa: \ker f \rightarrow G (kernels are always monomorphisms), which complete the short exact sequence 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 morphisms from \ker f to H and G / \ker f. If the sequence is right split (i.e., there is a morphism ''σ'' that maps G / \operatorname f to a -preimage of itself), then ''G'' is the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in ...
of the normal subgroup \operatorname \kappa and the subgroup \operatorname \sigma. If it is left split (i.e., there exists some \rho: G \rightarrow \operatorname f such that \rho \circ \kappa = \operatorname_), then it must also be right split, and \operatorname \kappa \times \operatorname \sigma is a
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
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 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 prototypical example of an abelian category is the category of a ...
(such as that of abelian groups), left splits and right splits are equivalent by the splitting lemma, and a right split is sufficient to produce a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
decomposition \operatorname \kappa \oplus \operatorname \sigma. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence 0 \rightarrow G / \operatorname f \rightarrow H \rightarrow \operatorname f \rightarrow 0. In the second isomorphism theorem, the product ''SN'' is the
join Join may refer to: * Join (law), to include additional counts or additional defendants on an indictment *In mathematics: ** Join (mathematics), a least upper bound of sets orders in lattice theory ** Join (topology), an operation combining two topo ...
of ''S'' and ''N'' in the
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. In this lattice, the join of two subgroups is the subgroup generated by their uni ...
of ''G'', while the intersection ''S'' ∩ ''N'' is the meet. The third isomorphism theorem is generalized by the
nine lemma right In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category In mathematics, an abelian category is a category in which morphisms and ...
to abelian categories and more general maps between objects.


Rings

The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
.


Theorem A (rings)

Let ''R'' and ''S'' be rings, and let ''φ'' : ''R'' → ''S'' be a ring homomorphism. Then: # The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of ''φ'' is an ideal of ''R'', # The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''φ'' is a subring of ''S'', and # The image of ''φ'' is isomorphic to the quotient ring ''R'' / ker(''φ''). In particular, if ''φ'' is surjective then ''S'' is isomorphic to ''R'' / ker(''φ'').


Theorem B (rings)

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'' =  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.


Theorem C (rings)

Let ''R'' be a ring, and ''I'' an ideal of ''R''. Then # If A is a subring of R such that I \subseteq A \subseteq R, then A/I is a subring of R/I. # Every subring of R/I is of the form A/I for some subring A of R such that I \subseteq A \subseteq R. # If J is an ideal of R such that I \subseteq J \subseteq R, then J/I is an ideal of R/I. # Every ideal of R/I is of the form J/I for some ideal J of R such that I \subseteq J \subseteq R. # If J is an ideal of R such that I \subseteq J \subseteq R, then the quotient ring (R/I)/(J/I) is isomorphic to R/J.


Theorem D (rings)

Let I be an ideal of R. The correspondence A\leftrightarrow A/I is an inclusion-preserving
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between the set of subrings A of R that contain I and the set of subrings of R/I. Furthermore, A (a subring containing I) is an ideal of R if and only if A/I is an ideal of R/I.


Modules

The statements of the isomorphism theorems for modules are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces (modules over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
) and
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s (modules over \mathbb) are special cases of these. For
finite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dist ...
vector spaces, all of these theorems follow from the rank–nullity theorem. In the following, "module" will mean "''R''-module" for some fixed ring ''R''.


Theorem A (modules)

Let ''M'' and ''N'' be modules, and let ''φ'' : ''M'' → ''N'' be a module homomorphism. Then: # The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of ''φ'' is a submodule of ''M'', # The
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''φ'' is a submodule of ''N'', and # The image of ''φ'' is isomorphic to the quotient module ''M'' / ker(''φ''). In particular, if ''φ'' is surjective then ''N'' is isomorphic to ''M'' / ker(''φ'').


Theorem B (modules)

Let ''M'' be a module, and let ''S'' and ''T'' be submodules of ''M''. Then: # The sum ''S'' + ''T'' =  is a submodule of ''M'', # The intersection ''S'' ∩ ''T'' is a submodule of ''M'', and # The quotient modules (''S'' + ''T'') / ''T'' and ''S'' / (''S'' ∩ ''T'') are isomorphic.


Theorem C (modules)

Let ''M'' be a module, ''T'' a submodule of ''M''. # If S is a submodule of M such that T \subseteq S \subseteq M, then S/T is a submodule of M/T. # Every submodule of M/T is of the form S/T for some submodule S of M such that T \subseteq S \subseteq M. # If S is a submodule of M such that T \subseteq S \subseteq M, then the quotient module (M/T)/(S/T) is isomorphic to M/S.


Theorem D (modules)

Let M be a module, N a submodule of M. There is a bijection between the submodules of M that contain N and the submodules of M/N. The correspondence is given by A\leftrightarrow A/N for all A\supseteq N. This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of M/N and the lattice of submodules of M that contain N).


Universal algebra

To generalise this to universal algebra, normal subgroups need to be replaced by
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
s. A congruence on an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
A is an equivalence relation \Phi\subseteq A \times A that forms a subalgebra of A \times A considered as an algebra with componentwise operations. One can make the set of equivalence classes A/\Phi into an algebra of the same type by defining the operations via representatives; this will be well-defined since \Phi is a subalgebra of A \times A. The resulting structure is the quotient algebra.


Theorem A (universal algebra)

Let f:A \rightarrow B be an algebra homomorphism. Then the image of f is a subalgebra of B, the relation given by \Phi:f(x)=f(y) (i.e. the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of f) is a congruence on A, and the algebras A/\Phi and \operatorname f are isomorphic. (Note that in the case of a group, f(x)=f(y)
iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
f(xy^) = 1, so one recovers the notion of kernel used in group theory in this case.)


Theorem B (universal algebra)

Given an algebra A, a subalgebra B of A, and a congruence \Phi on A, let \Phi_B = \Phi \cap (B \times B) be the trace of \Phi in B and \Phi=\ the collection of equivalence classes that intersect B. Then # \Phi_B is a congruence on B, # \ \Phi is a subalgebra of A/\Phi, and # the algebra \Phi is isomorphic to the algebra B/\Phi_B.


Theorem C (universal algebra)

Let A be an algebra and \Phi, \Psi two congruence relations on A such that \Psi \subseteq \Phi. Then \Phi/\Psi = \ = \Psi \circ \Phi \circ \Psi^ is a congruence on A/\Psi, and A/\Phi is isomorphic to (A/\Psi)/(\Phi/\Psi).


Theorem D (universal algebra)

Let A be an algebra and denote \operatornameA the set of all congruences on A. The set \operatornameA is a complete lattice ordered by inclusion. If \Phi\in\operatornameA is a congruence and we denote by \left Phi,A\times A\rightsubseteq\operatornameA the set of all congruences that contain \Phi (i.e. \left Phi,A\times A\right/math> is a principal filter in \operatornameA, moreover it is a sublattice), then the map \alpha:\left Phi,A\times A\rightto\operatorname(A/\Phi),\Psi\mapsto\Psi/\Phi is a lattice isomorphism.


Note


References

* Emmy Noether, ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', Mathematische Annalen 96 (1927) pp. 26–61 * Colin McLarty, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". ''The Architecture of Modern Mathematics: Essays in history and philosophy'' (edited by
Jeremy Gray Jeremy John Gray (born 25 April 1947) is an English mathematician primarily interested in the history of mathematics. Biography Gray studied mathematics at Oxford University from 1966 to 1969, and then at Warwick University, obtaining his Ph.D ...
and José Ferreirós), Oxford University Press (2006) pp. 211–35. * * Paul M. Cohn, ''Universal algebra'', Chapter II.3 p. 57 * * * * * * * * * {{citation , author = Joseph J. Rotman , title=Advanced Modern Algebra , publisher=Prentice Hall , edition= 2 , year=2003 , isbn=0130878685