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# Isomorphism theorem

In mathematics, the isomorphism theorems are 3 theorems that apply broadly in the realm of universal algebra.

### Groups

First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of factor groups (also called quotient groups).

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, and the factor group G/K is isomorphic to the image of f.

Second Isomorphism Theorem. Let N be a normal subgroup of the group G, and let S be any subgroup. The intersection N ∩ S of N and S is a normal subgroup of S, N is a normal subgroup of the join NS of N and S, and S/(N ∩ S) is isomorphic to SN/N.

Third 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.

### 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[?]".

The isomorphism theorems are also valid for rings, ring homomorphisms and ideals. One has to replace the term "group" by "ring", "subgroup" and "normal subgroup" by "ideal", and "factor group" by "factor ring".

The notation for the join in both these cases is "S + N" instead of "S'N".

We also need to mention the isomorphism theorems for topological vector spaces, Banach algebras etc.

### General

To generalise this to universal algebra, normal subgroups need to be replaced with congruences.

Finish later -- time for bed.

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