Isomorphism theorems — Safekipedia Discoverer

In mathematics, especially in abstract algebra, the isomorphism theorems—also called Noether's isomorphism theorems—are important ideas that help us understand how different mathematical objects relate to each other. These theorems explain the links between quotients, homomorphisms, and subobjects. They show how we can break down complex structures into simpler parts and still keep track of their relationships.

These theorems are useful in many areas of math. They apply to groups, rings, vector spaces, modules, Lie algebras, and other types of algebraic structures. This means whether you are studying shapes, numbers, or more abstract ideas, these theorems can help you see patterns and connections.

In universal algebra, the isomorphism theorems can even be expanded to work with more general systems called algebras, using the idea of congruences. This makes them a powerful tool for mathematicians working in many different fields.

History

The isomorphism theorems were first explained in detail by Emmy Noether in 1927. She wrote about them in a paper called Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in a math journal named Mathematische Annalen. Later, in 1930, another mathematician named B. L. van der Waerden wrote a very important book called Moderne Algebra. This book helped make these theorems well-known and easy for others to understand. It also helped organize the study of groups, rings, and fields, which are important parts of algebra.

Groups

The isomorphism theorems for groups help us understand relationships between groups, their subgroups, and quotients. These theorems are important tools in abstract algebra.

Theorem A (groups)

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

The kernel of f is a normal subgroup of G,
The image of f is a subgroup of H, and
The image of f is isomorphic to the quotient group G / ker f.

If f is surjective, then H is isomorphic to G / ker f. This is known as the first isomorphism theorem.

Theorem B (groups)

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

The product SN is a subgroup of G,
N is a normal subgroup of SN,
The intersection S ∩ N is a normal subgroup of S, and
The quotient groups (SN)/N and S/(S ∩ N) are isomorphic.

This theorem is sometimes called the second isomorphism theorem.

Diagram for theorem B4. The two quotient groups (dotted) are isomorphic.

Theorem C (groups)

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

If K is a subgroup of G such that N ⊆ K ⊆ G, then G/N has a subgroup isomorphic to K/N.
Every subgroup of G/N is of the form K/N for some subgroup K of G such that N ⊆ K ⊆ G.
If K is a normal subgroup of G such that N ⊆ K ⊆ G, then G/N has a normal subgroup isomorphic to K/N.
Every normal subgroup of G/N is of the form K/N for some normal subgroup K of G such that N ⊆ K ⊆ G.
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.

The last statement is known as the third isomorphism theorem.

Theorem D (groups)

Let G be a group and N a normal subgroup of G. There is a bijective correspondence between the set of subgroups of G containing N and the set of all subgroups of G/N. Under this correspondence, normal subgroups correspond to normal subgroups. This theorem is sometimes called the correspondence theorem or the lattice theorem.

The first isomorphism theorem can also be described using category theory. It shows that the category of groups is factorizable, meaning that normal epimorphisms and monomorphisms can be used to break down morphisms into simpler parts.

Comparison of the names of the group isomorphism theorems
Comment	Author	Theorem A	Theorem B	Theorem C
No "third" theorem	Jacobson	Fundamental theorem of homomorphisms	(Second isomorphism theorem)	"often called the first isomorphism theorem"
	van der Waerden, Durbin	Fundamental theorem of homomorphisms	First isomorphism theorem	Second isomorphism theorem
	Knapp	(No name)	Second isomorphism theorem	First isomorphism theorem
	Grillet	Homomorphism theorem	Second isomorphism theorem	First isomorphism theorem
Three numbered theorems	(Other convention per Grillet)	First isomorphism theorem	Third isomorphism theorem	Second isomorphism theorem
	Rotman	First isomorphism theorem	Second isomorphism theorem	Third isomorphism theorem
	Fraleigh	Fundamental homomorphism theorem or first isomorphism theorem	Second isomorphism theorem	Third isomorphism theorem
	Dummit & Foote	First isomorphism theorem	Second or Diamond isomorphism theorem	Third isomorphism theorem
No numbering	Milne	Homomorphism theorem	Isomorphism theorem	Correspondence theorem
No numbering	Scott	Homomorphism theorem	Isomorphism theorem	Freshman theorem

Rings

The statements of the theorems for rings are similar, with the notion of a normal subgroup replaced by the notion of an ideal.

These theorems help us understand how smaller parts of a ring relate to the whole ring. They show connections between parts of a ring, special subsets called ideals, and quotients, which are like simplifying the ring by grouping similar elements together.

Main article: Isomorphism theorems

Modules

The isomorphism theorems for modules are easy to understand because you can create a quotient module from any submodule. These theorems also apply to vector spaces, which are modules over a field, and to abelian groups, which are modules over the whole numbers (Z). For finite-dimensional vector spaces, these ideas come from the rank–nullity theorem.

These theorems help us see how different parts of a module relate to each other. For example, they show how the part left after removing some elements (the kernel) looks like another part (the image). They also explain how combining or comparing different submodules works, and how submodules of a bigger module relate to submodules of a smaller one created by "dividing out" part of the bigger module.

Universal algebra

In mathematics, especially in a part called abstract algebra, there are special rules called isomorphism theorems. They help us understand how different math structures relate to each other. These theorems work for many types of math objects, like groups, rings, and vector spaces.

To make these ideas work for more general math structures, we use something called "congruence relations" instead of normal subgroups. A congruence is a way to group elements of an algebra that keeps the algebra's operations consistent. This lets us create new algebras from these groups of elements, called quotient algebras. The isomorphism theorems tell us when these new algebras are essentially the same as other algebras, even if they look different at first glance.