Isomorphism theorem

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In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms.

Contents 1 History 2 Groups 2.1 First isomorphism theorem 2.2 Second isomorphism theorem (also known as the third isomorphism theorem) 2.3 Third isomorphism theorem (also known as the second isomorphism theorem) 3 Rings and modules 4 General 4.1 First Isomorphism Theorem 4.2 Second Isomorphism Theorem 4.3 Third Isomorphism Theorem 5 References 6 See also 7 External links

[edit] 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 now-traditional groups-rings-fields 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.

[edit] Groups

First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of quotient groups (also called factor groups). All three involve "modding out" by a normal subgroup.

[edit] First isomorphism theorem

First isomorphism theorem

If G and H are groups and f is a homomorphism from G to H, then the kernel K of f is a normal subgroup of G, the image of f is a subgroup of H, and the quotient group G /K is isomorphic to the image of f.

If

then

In particular, if f is surjective and the kernel is trivial, then G is isomorphic to H.

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 structure 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 π-preimage 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 .

[edit] Second isomorphism theorem (also known as the third isomorphism theorem)

Let H and K be subgroups of the group G, and assume H is a subgroup of the normalizer of K. Then the join HK of H and K is a subgroup of G, K is a normal subgroup of HK, H ∩K is a normal subgroup of H, and HK /K is isomorphic to H /(H ∩K).

If

then

[edit] Third isomorphism theorem (also known as the second isomorphism theorem)

If M and N are normal subgroups of G such that M is contained in N, then M is a normal subgroup of N, N /M is a normal subgroup of G /M, and (G /M) /(N /M) is isomorphic to G /N.

If

then

This is generalized by the nine lemma to abelian categories and more general maps between objects.

[edit] Rings and modules

The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field). One has to replace the term "group" by "R-module", "subgroup" and "normal subgroup" by "submodule", and "factor group" by "factor module".

For vector spaces, the first isomorphism theorem goes by the name of rank-nullity theorem.

The isomorphism theorems are also valid for rings, ring homomorphisms and ideals. One has to replace the term "group" by "ring", "subgroup" by "subring" and "normal subgroup" by "ideal", and "factor group" by "factor ring".

The notation for the join in both these cases is "H + K" instead of "HK".

[edit] 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 A x A (the latter with the coordinate-wise 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 well-defined since Φ is a subalgebra of A x A.

[edit] First Isomorphism Theorem

If A and B are algebras, and f is a homomorphism from A to B, 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.

[edit] 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 A x A) with B x B. 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.

[edit] 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 Ψ-equivalence class of a), and A/Φ is isomorphic to (A/Ψ)/Θ.

[edit] References

• Emmy Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Mathematische Annalen 96 (1927) p. 26-61
• 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.

[edit] 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

[edit] External links

• First isomorphism theorem on PlanetMath. Proof of first isomorphism theorem on PlanetMath
• Second isomorphism theorem on PlanetMath. Proof of second isomorphism theorem on PlanetMath
• Third isomorphism theorem on PlanetMath. Proof of third isomorphism theorem on PlanetMath