fundamental homomorphism theorem
The following theorem is also true for rings (with ideals instead of normal subgroups) or modules (with submodules
instead of normal subgroups).
theorem 1.
Let G,H be groups, f:G→H a homomorphism, and let N be a normal subgroup of G contained in ker(f). Then there exists a unique homomorphism h:G/N→H so that h∘φ=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 injective.
Proof.
We’ll first show the uniqueness. Let h1,h2:G/N→H functions such that h1∘φ=h2∘φ. For an element y in G/N there exists an element x in G such that φ(x)=y, so we have
h1(y)=(h1∘φ)(x)=(h2∘φ)(x)=h2(y) |
for all y∈G/N, thus h1=h2.
Now we define h:G/N→H,h(gN)=f(g)∀g∈G. We must check that the definition is of the given representative; so let gN=kN, or k∈gN. Since N is a subset of ker(f), g-1k∈N implies g-1k∈ker(f), hence f(g)=f(k). Clearly h∘φ=f.
Since x∈ker(f) if and only if h(xN)=1H, we have
ker(h)={xN∣x∈ker(f)}=ker(f)/N. |
∎
A consequence of this is: If f:G→H is onto with ker(f)=N, then G/N and H are isomorphic.
Title | fundamental homomorphism theorem![]() |
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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 |