# Fundamental theorem on homomorphisms

﻿
Fundamental theorem on homomorphisms

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

The homomorphism theorem is used to prove the isomorphism theorems.

Group theoretic version

Given two groups "G" and "H" and a group homomorphism "f" : "G"&rarr;"H", let "K" be a normal subgroup in "G" and &phi; the natural surjective homomorphism "G"&rarr;"G"/"K". If "K" &sub; ker("f") then there exists a unique homomorphism "h":"G"/"K"&rarr;"H" such that "f" = "h" &phi;.

The situation is described by the following commutative diagram:

By setting "K" = ker("f") we immediately get the first isomorphism theorem.

Other versions

Similar theorems are valid for monoids, vector spaces, modules, and rings.

ee also

* Quotient category

* [http://planetmath.org/encyclopedia/FundamentalHomomorphismTheorem.html A proof at planetmath]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Fundamental theorem — In mathematics, there are a number of fundamental theorems for different fields. The names are mostly traditional; so that for example the fundamental theorem of arithmetic is basic to what would now be called number theory. Theorems may be… …   Wikipedia

• Seifert–van Kampen theorem — In mathematics, the Seifert–van Kampen theorem of algebraic topology, sometimes just called van Kampen s theorem, expresses the structure of the fundamental group of a topological space X, in terms of the fundamental groups of two open, path… …   Wikipedia

• Atiyah–Singer index theorem — In the mathematics of manifolds and differential operators, the Atiyah–Singer index theorem states that for an elliptic differential operator on a compact manifold, the analytical index (closely related to the dimension of the space of solutions) …   Wikipedia

• Whitehead theorem — In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between topological spaces X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence provided X and Y are… …   Wikipedia

• Stone's representation theorem for Boolean algebras — In mathematics, Stone s representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of… …   Wikipedia

• Kernel (algebra) — In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also… …   Wikipedia

• List of group theory topics — Contents 1 Structures and operations 2 Basic properties of groups 2.1 Group homomorphisms 3 Basic types of groups …   Wikipedia

• List of mathematics articles (F) — NOTOC F F₄ F algebra F coalgebra F distribution F divergence Fσ set F space F test F theory F. and M. Riesz theorem F1 Score Faà di Bruno s formula Face (geometry) Face configuration Face diagonal Facet (mathematics) Facetting… …   Wikipedia

• Glossary of group theory — A group ( G , •) is a set G closed under a binary operation • satisfying the following 3 axioms:* Associativity : For all a , b and c in G , ( a • b ) • c = a • ( b • c ). * Identity element : There exists an e ∈ G such that for all a in G , e •… …   Wikipedia

• List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia