Isomorphism theorem

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.

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1 History
2 Groups
3 Rings and modules
4 General
5 References
6 See also
7 External links

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

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.

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.

$G, H \text{ are groups}\;$

$f: G \to H, f \text{ is a homomorphism}\;$

then

$\operatorname{Ker}(f) \triangleleft G$

$\operatorname{Im}(f) \leq H$

$G/\operatorname{Ker}(f) \cong \operatorname{Im}(f)$

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).

$H,K \leq G \,$

$H \leq \operatorname{N}_G(K)$

then

$HK \leq G \,$

$K \triangleleft HK$

$H \cap K \triangleleft H$

$HK/K \cong H/(H \cap K)$

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.

$M,N \triangleleft G$

$M \subseteq N$

then

$M \triangleleft N$

$N/M \triangleleft G/M$

$(G/M)/(N/M) \cong G/N$

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

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".

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.

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.

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/ $Φ B$ .

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/ $Ψ$ )/ $Θ$ .

References

Emmy Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Mathematische Annalen 96 (1927) p. 26-61
Colin McLarty (edited by Jeremy Gray and José Ferreirós), The Architecture of Modern Mathematics: Essays in history and philosophy - Emmy Noether’s ‘Set Theoretic’ Topology: From Dedekind to the rise of functors, Oxford University Press (2006) p. 211–35.

External links

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Contents

History

Groups

First isomorphism theorem

Second isomorphism theorem (also known as the third isomorphism theorem)

Third isomorphism theorem (also known as the second isomorphism theorem)

Rings and modules

General

First Isomorphism Theorem

Second Isomorphism Theorem

Third Isomorphism Theorem

References

See also

External links