Isomorphism theorem

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(Redirected from Second 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.

[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

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.

[edit] Statement of the theorems

[edit] 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).

[edit] 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.

[edit] 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.

[edit] 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 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 $\iota \circ \pi$ , 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.