 Isomorphism theorem

In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.
Contents
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 Algebra, the first abstract algebra textbook that took the nowtraditional groupsringsfields approach to the subject. Van der Waerden credited lectures by Noether on group theory 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 state the three isomorphism theorems in the context of groups. Note that some sources switch the numbering of the second and third theorems.^{[1]} Sometimes, the lattice theorem is referred to as the fourth isomorphism theorem or the correspondence theorem.
Statement of the theorems
First isomorphism theorem
Let G and H be groups, and let φ: G → H be a homomorphism. Then:
 The kernel of φ is a normal subgroup of G,
 The image of φ is a subgroup of H, and
 The image of φ is isomorphic to the quotient group G / ker(φ).
In particular, if φ is surjective 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 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 fact that the category of groups is (normal epi, mono)factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram 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 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 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 to H and .
If the sequence is right split (i. e., there is a morphism σ that maps to a πpreimage of itself), then G is the semidirect 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 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 (such as the abelian groups), left splits and right splits are equivalent by the splitting lemma, 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 of G, while the intersection S ∩ N is the meet.
The third isomorphism theorem is generalized by the nine lemma 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 are similar, with the notion of a normal subgroup replaced by the notion of an ideal.
First isomorphism theorem
Let R and S be rings, and let φ: R → S be a ring homomorphism. Then:
 The kernel of φ is an ideal of R,
 The image 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).
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 are particularly simple, since it is possible to form a quotient module from any submodule. The isomorphism theorems for vector spaces and abelian groups are special cases of these. For vector spaces, all of these theorems follow from the ranknullity theorem.
For all of the following theorems, the word “module” will mean “Rmodule”, 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 of φ is a submodule of M,
 The image 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).
Second isomorphism theorem
Let M be a module, and let S and T be submodules of M. Then:
 The sum S + T = {s + t  s ∈ S, t ∈ T} is a submodule of M,
 The intersection S ∩ T is a submodule of S, and
 The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic.
Third isomorphism theorem
Let M be a module. Let S and T be submodules of M, with
 T ⊆ S ⊆ M.
Then
 The quotient S / T is a submodule of the quotient M / T, and
 The quotient (M / T) / (S / T) is isomorphic to M / S.
General
To generalise this to universal algebra, normal subgroups need to be replaced by congruences.
Briefly, if A is an algebra, a congruence on A is an equivalence relation Φ on A which is a subalgebra when considered as a subset of (the latter with the coordinatewise operation structure). One can make the set of equivalence classes A / Φ into an algebra of the same type by defining the operations via representatives; this will be welldefined since Φ is a subalgebra of .
First Isomorphism Theorem
If A and B are algebras, and f is a homomorphism , then the equivalence relation Φ on A defined by a∼b if and only if f(a) = f(b) is a congruence on A, and the algebra A / Φ is isomorphic to the image of f, which is a subalgebra of B.
Second Isomorphism Theorem
Given an algebra A, a subalgebra B of A, and a congruence Φ on A, we let [B]Φ be the subset of A / Φ determined by all congruence classes that contain an element of B, and we let Φ_{B} be the intersection of Φ (considered as a subset of ) with . Then [B]Φ is a subalgebra of A / Φ, Φ_{B} is a congruence on B, and the algebra [B]Φ is isomorphic to the algebra B / Φ_{B}.
Third Isomorphism Theorem
Let A be an algebra, and let Φ and Ψ be two congruence relations on A, with Ψ contained in Φ. Then Φ determines a congruence Θ on A / Ψ defined by [a]∼[b] if and only if a and b are equivalent modulo Φ (where [a] represents the Ψequivalence class of a), and A / Φ is isomorphic to (A / Ψ) / Θ.
See also
 Butterfly lemma, sometimes called the fourth isomorphism theorem
 Lattice theorem, sometimes called the fourth isomorphism theorem
 Splitting lemma, which refines the first isomorphism theorem for split sequences
Notes
 ^ Jacobson (2009), p. 101, use "first" for the isomorphism of the modules (S + T) / T and S / (S ∩ T), and "second" for (M / T) / (S / T) and M / S.
References
 Emmy Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl und Funktionenkörpern, Mathematische Annalen 96 (1927) p. 2661
 Colin McLarty, 'Emmy Noether’s ‘Set Theoretic’ Topology: From Dedekind to the rise of functors' in The Architecture of Modern Mathematics: Essays in history and philosophy (edited by Jeremy Gray and José Ferreirós), Oxford University Press (2006) p. 211–35.
 Jacobson, Nathan (2009), Basic algebra, 2 (2nd ed.), Dover, ISBN 9780486471877
External links
Categories: Theorems in algebra
 Isomorphism theorems
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