Isomorphism theorem

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 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.

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. Another variation encountered in the literature, particularly in Van der Waerden's Algebra, is to call first isomorphism theorem the Fundamental Homomorphism Theorem and consequently to decrement the numbering of the remaining isomorphism theorems by one. Finally, in the most extensive numbering scheme, the lattice theorem (also known as the correspondence theorem) is sometimes referred to as the fourth isomorphism theorem.

First isomorphism theorem

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

1. The kernel of φ is a normal subgroup of G,
2. The image of φ is a subgroup of H, and
3. 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 following hold:

1. The product S N is a subgroup of G ,
2. The intersection S ∩ N is a normal subgroup of S , and
3. The quotient groups ( S N ) / 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.

This isomorphism theorem has been called the "diamond theorem" due to the shape of the resulting subgroup lattice with S N at the top, S ∩ N at the bottom and with N and S to the sides. It has even been called the "parallelogram rule" (by analogy with the parallelogram rule for vectors) because in the resulting subgroup lattice the two sides assumed to represent the quotient groups ( S N ) / N and S / ( S ∩ N ) are "equal" in the sense of isomorphism.

An example of an application of the second isomorphism theorem is with projective linear groups. Setting G = G L 2 ( C ) (the group of invertible 2x2 complex matrices), S = S L 2 ( C ) (subgroup of matrices with determinant 1), and N the normal subgroup generated by C ∗ ⋅ Id 2 = { ( a 0 0 a ) : a ∈ C ∗ } . Then, the second isomorphism theorem states that

P G L 2 ( C ) := G L 2 ( C ) / C ∗ ⋅ Id 2 ≅ S L 2 ( C ) / { ± Id 2 } =: P S L 2 ( C )

since { ± Id 2 } = S ∩ N and G L 2 ( C ) = S N .

Third isomorphism theorem

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

1. If K is a subgroup of G such that N ⊆ K ⊆ G , then K / N is a subgroup of G / N .
2. Every subgroup of G / N is of the form K / N , for some subgroup K of G such that N ⊆ K ⊆ G .
3. If K is a normal subgroup of G such that N ⊆ K ⊆ G , then K / N is a normal subgroup of G / N .
4. Every normal subgroup of G / N is of the form K / N , for some normal subgroup K of G such that N ⊆ K ⊆ G .
5. If K is a normal subgroup of G such that N ⊆ K ⊆ G , then the quotient group ( G / N ) / ( K / N ) is isomorphic to G / K .

W.R. Scott calls it the "Freshman theorem" because the result simply follows by "cancellation" of N.

Discussion

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 ι ∘ π , 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 κ : ker ⁡ f → 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 / ker ⁡ f to a π-preimage of itself), then G is the semidirect product of the normal subgroup im κ and the subgroup im σ . If it is left split (i. e., there exists some ρ : G → ker ⁡ f such that ρ ∘ κ = id ker ⁡ f ), then it must also be right split, and im κ × im σ 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 im κ ⊕ im σ . In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence 0 → G / ker ⁡ f → H → coker ⁡ f → 0 .

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.

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:

1. The kernel of φ is an ideal of R,
2. The image of φ is a subring of S, and
3. 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:

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

Third isomorphism theorem

Let R be a ring, and B an ideal of R. Then

1. If A is a subring of R such that B ⊆ A ⊆ R , then A / B is a subring of R / B .
2. Every subring of R / B is of the form A / B , for some subring A of R such that B ⊆ A ⊆ R .
3. If A is an ideal of R such that B ⊆ A ⊆ R , then A / B is an ideal of R / B .
4. Every ideal of R / B is of the form A / B , for some ideal A of R such that B ⊆ A ⊆ R .
5. If A is an ideal of R such that B ⊆ A ⊆ R , then 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 finite-dimensional vector spaces, all of these theorems follow from the rank-nullity theorem.

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 module homomorphism. Then:

1. The kernel of φ is a submodule of M,
2. The image of φ is a submodule of N, and
3. 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:

1. The sum S + T = {s + t | s ∈ S, t ∈ T} is a submodule of M,
2. The intersection S ∩ T is a submodule of M, and
3. The quotient modules (S + T) / T and S / (S ∩ T) are isomorphic.

Third isomorphism theorem

Let M be a module, T a submodule of M.

1. If S is a submodule of M such that T ⊆ S ⊆ M , then S / T is a submodule of M / T .
2. Every submodule of M / T is of the form S / T , for some submodule S of M such that T ⊆ S ⊆ M .
3. If S is a submodule of M such that T ⊆ S ⊆ M , then the quotient module ( M / T ) / ( S / T ) is isomorphic to M / S .

General

To generalise this to universal algebra, normal subgroups need to be replaced by congruences.

A congruence on an algebra A is an equivalence relation Φ which is a subalgebra of A × A endowed with the component-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 × A . The resulting structure is the quotient algebra.

First Isomorphism Theorem

Let f : A → B be an algebra homomorphism. Then the image of f is a subalgebra of B , the relation given by Φ : f ( x ) = f ( y ) (i.e. the kernel of f) is a congruence on A , and the algebras   A / Φ   and im ⁡ f are isomorphic. (Note that in the case of a group, f(x) = f(y) iff f(xy⁻¹) = 1, so one recovers the notion of kernel used in group theory in this case.)

Second Isomorphism Theorem

Given an algebra A , a subalgebra B of A , and a congruence Φ on A , let Φ B = Φ ∩ ( B × B ) be the trace of Φ in B and [ B ] Φ = { K ∈ A / Φ : K ∩ B ≠ ∅ } the collection of equivalence classes that intersect B .

Then (i) Φ B is a congruence on B , (ii)   [ B ] Φ is a subalgebra of A / Φ , and (iii) the algebra [ B ] Φ is isomorphic to the algebra B / Φ B .

Third Isomorphism Theorem

Let A be an algebra and Φ , Ψ two congruence relations on A such that Ψ ⊆ Φ . Then Φ / Ψ = { ( [ a ′ ] Ψ , [ a ″ ] Ψ ) : ( a ′ , a ″ ) ∈ Φ } = [   ] Ψ ∘ Φ ∘ [   ] Ψ − 1 is a congruence on A / Ψ , and A / Φ is isomorphic to ( A / Ψ ) / ( Φ / Ψ ) .