Isomorphism theorems

The isomorphism theorems relate homomorphisms, substructures, and quotient structures by identifying when different constructions produce isomorphic objects.

Group homomorphism: A map between groups that preserves the group operation. The theorems describe how homomorphisms connect kernels, images, subgroups, and quotients.
First isomorphism theorem: The theorem that a group modulo a homomorphism's kernel is isomorphic to its image. It is the basic quotient-image correspondence behind the family.
Fundamental theorem of homomorphisms: A common name for the first isomorphism theorem in algebraic settings. It lets a homomorphism's image be computed as a quotient by its kernel.
Sylow theorems: Theorems describing the existence, conjugacy, and number of Sylow subgroups in finite groups. Unlike the isomorphism theorems, they constrain subgroup counts and conjugacy.
Kernel of a group homomorphism: The subgroup of elements mapped to the identity by a group homomorphism. The first theorem identifies the quotient by this subgroup with the homomorphism's image.
Second isomorphism theorem: The theorem relating a subgroup and a normal subgroup through their product and intersection. It expresses a quotient of a product subgroup as a quotient of the original subgroup.
Chinese remainder theorem: A theorem giving conditions under which a quotient by a product of ideals decomposes into a product of quotients. Its ring-theoretic form uses quotient isomorphisms to decompose residue structures.
Jordan–Hölder theorem: The theorem that finite composition series of a group have the same simple factors up to order. It classifies composition factors, while the isomorphism theorems relate particular quotients and subgroups.
Image of a group homomorphism: The subgroup consisting of all values attained by a group homomorphism. It is the target of the first theorem's quotient-to-image isomorphism.
Third isomorphism theorem: The theorem identifying a quotient by nested normal subgroups with a quotient of the larger quotient. It shows how successive quotienting collapses to one quotient.

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