fundamental homomorphism theorem

The following theorem is also true for rings (with ideals instead of normal subgroupsMathworldPlanetmath) or modules (with submodulesMathworldPlanetmath instead of normal subgroups).

theorem 1.

Let G,H be groups, f:GH a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and let N be a normal subgroup of G contained in ker(f). Then there exists a unique homomorphism h:G/NH so that hφ=f, where φ denotes the canonical homomorphism from G to G/N.

Furthermore, if f is onto, then so is h; and if ker(f)=N, then h is injectivePlanetmathPlanetmath.


We’ll first show the uniqueness. Let h1,h2:G/NH functions such that h1φ=h2φ. For an element y in G/N there exists an element x in G such that φ(x)=y, so we have


for all yG/N, thus h1=h2.

Now we define h:G/NH,h(gN)=f(g)gG. We must check that the definition is of the given representative; so let gN=kN, or kgN. Since N is a subset of ker(f), g-1kN implies g-1kker(f), hence f(g)=f(k). Clearly hφ=f.

Since xker(f) if and only if h(xN)=1H, we have


A consequence of this is: If f:GH is onto with ker(f)=N, then G/N and H are isomorphic.

Title fundamental homomorphism theoremMathworldPlanetmath
Canonical name FundamentalHomomorphismTheorem
Date of creation 2013-03-22 15:35:06
Last modified on 2013-03-22 15:35:06
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Theorem
Classification msc 20A05