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 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.

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:

  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:

  1. The product SN is a subgroup of G,
  2. The intersection S ∩ N is a normal subgroup of S, and
  3. 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

  1. The quotient N / K is a normal subgroup of the quotient G / K, and
  2. 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 \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 / \ker\, f to a π-preimage of itself), then G is the semidirect product of the normal subgroup \operatorname{im}\, \kappa and the subgroup \operatorname{im}\, \sigma. If it is left split (i. e., there exists some \rho: G \rightarrow \ker\, f such that \rho \circ \kappa = \operatorname{id}_{\ker\, f}), then it must also be right split, and \operatorname{im}\, \kappa \times \operatorname{im}\, \sigma 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 \operatorname{im}\, \kappa \oplus \operatorname{im}\, \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 / \ker\, f \rightarrow H \rightarrow \operatorname{coker}\, f \rightarrow 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. 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:

  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. Let A and B be ideals of R, with

B ⊆ A ⊆ R.

Then

  1. The set A / B is an ideal of the quotient R / B, and
  2. 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 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 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 S, and
  3. 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

  1. The quotient S / T is a submodule of the quotient M / T, and
  2. 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 \Phi on A which is a subalgebra when considered as a subset of A \times A (the latter with the coordinate-wise operation structure). 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.

First Isomorphism Theorem

If A and B are algebras, and f is a homomorphism f:A \rightarrow B, then the equivalence relation \Phi on A defined by a \sim b if and only if f(a)=f(b) is a congruence on A, and the algebra A/\Phi 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 \Phi on A, we let [B]\Phi be the subset of A/\Phi determined by all congruence classes that contain an element of B, and we let \Phi_B be the intersection of \Phi (considered as a subset of A \times A) with B \times B. Then [B]\Phi is a subalgebra of A/\Phi, \Phi_B is a congruence on B, and the algebra [B]\Phi is isomorphic to the algebra B/\Phi_B.

Third Isomorphism Theorem

Let A be an algebra, and let \Phi and \Psi be two congruence relations on A, with \Psi contained in \Phi. Then \Phi determines a congruence \Theta on A/\Psi defined by [a] \sim [b] if and only if a and b are equivalent modulo \Phi (where [a] represents the \Psi-equivalence class of a), and A/\Phi is isomorphic to (A/\Psi)/\Theta.

See also

Notes

  1. ^ 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

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