Overview of the first isomorphism theorem第一同型定理だいいちどうけいていり

The central idea of the first isomorphism theorem第一同型定理だいいちどうけいていり is simple. If elements that collapse to the same value under a homomorphism are first identified, the remaining structure is the same as the image. This result is also called the first homomorphism theorem; this material consistently uses “first isomorphism theorem.”

This viewpoint is common to groups群ぐん, rings環かん, and linear algebra.

Linear algebra is included only as an analogy. The proofs on this page use only kernels, images, and quotients already introduced for groups and rings.

1Form of the first isomorphism同型どうけい theorem

For a group homomorphism群準同型ぐんじゅんどうけい $$\phi :G\to H$$\varphi:G\to H,

holds.

For a ring homomorphism環準同型かんじゅんどうけい $$\phi :R\to S$$\varphi:R\to S, the same form holds:

The formula has the same shape. The difference is that in group theory the kernel核かく is a normal subgroup正規部分群せいきぶぶんぐん, while in ring theory the kernel核かく is an idealイデアル.

2Why this happens

For a group homomorphism $$\phi :G\to H$$\varphi:G\to H,

Thus two elements have the same image exactly when they lie in the same left coset $$g\ker \phi$$g\ker\varphi. Therefore the map sending the coset剰余類じょうよるい

to

is independent of the choice of representative.

For a ring homomorphism $$\phi :R\to S$$\varphi:R\to S,

Thus two elements have the same image exactly when they lie in the same additive coset $$r+\ker \phi$$r+\ker\varphi. Therefore $$r+\ker \phi \mapsto \phi \left(r\right)$$r+\ker\varphi\mapsto\varphi(r) is also independent of the choice of representative.

3Correspondence with linear algebra

For a linear map $$T:V\to W$$T:V\to W, the kernel核かく is

and the image像ぞう is

Collapsing the kernel核かく directions leaves the degrees of freedom that become the image像ぞう. This intuition connects to rank, degeneracy, and quotient spaces.

This correspondence is only a preview; the first isomorphism theorem is not being proved by using theorems about linear maps.

4What changes and what is preserved

In the first isomorphism theorem, differences inside the kernel核かく are collapsed. What changes is the fineness with which elements are distinguished. What is preserved is the structure actually observable through the homomorphism, namely the image.

5Proof supplement: proof証明しょうめい of the first isomorphism同型どうけい theorem

Let $$\phi :G\to H$$\varphi:G\to H be a group homomorphism群準同型ぐんじゅんどうけい. The first isomorphism theorem第一同型定理だいいちどうけいていり states that

We first prove the group case and then prove the ring case.

The kernel of a group homomorphism is a normal subgroup, so $$G/\ker \phi$$G/\ker\varphi is defined as a quotient group.

Define the map

First show that it is well-defined. If $$g\ker \phi =g^{\prime }\ker \phi$$g\ker\varphi=g'\ker\varphi, then $$g^{-1}{g}^{\prime }\in \ker \phi$$g^{-1}g'\in\ker\varphi. Hence

and $$\phi \left(g{)}^{-1}\phi \right(g^{\prime })=e$$\varphi(g)^{-1}\varphi(g')=e, so $$\phi \left(g\right)=\phi \left(g^{\prime }\right)$$\varphi(g)=\varphi(g').

Next, $$\mathrm{\Phi }$$\Phi is a homomorphism準同型じゅんどうけい:

Surjectivity follows from the definition定義ていぎ of the image像ぞう. For injectivity, apply the previously proved criterion that a homomorphism is injective when its kernel contains only the identity. Indeed, if $$\mathrm{\Phi }\left(g\ker \phi \right)=e$$\Phi(g\ker\varphi)=e, then $$\phi \left(g\right)=e$$\varphi(g)=e, so $$g\in \ker \phi$$g\in\ker\varphi and therefore $$g\ker \phi =\ker \phi$$g\ker\varphi=\ker\varphi.

Thus $$\mathrm{\Phi }$$\Phi is an isomorphism同型どうけい, and $$G/\ker \phi \cong \mathrm{Im}\phi$$G/\ker\varphi\cong\operatorname{Im}\varphi.

Next let $$\phi :R\to S$$\varphi:R\to S be a ring homomorphism. Its kernel is an ideal of $$R$$R, so $$R/\ker \phi$$R/\ker\varphi is defined as a quotient ring. Consider

If $$r+\ker \phi =r^{\prime }+\ker \phi$$r+\ker\varphi=r'+\ker\varphi, then $$r-r^{\prime }\in \ker \phi$$r-r'\in\ker\varphi, so

Thus $$\mathrm{\Psi }$$\Psi is well-defined. By the definitions of addition and multiplication on residue classes,

so $$\mathrm{\Psi }$$\Psi is a ring homomorphism. Surjectivity follows from the definition of the image. If $$\mathrm{\Psi }(r+\ker \phi )=\mathrm{\Psi }(r^{\prime }+\ker \phi )$$\Psi(r+\ker\varphi)=\Psi(r'+\ker\varphi), then $$\phi (r-r^{\prime })=0$$\varphi(r-r')=0, so $$r-r^{\prime }\in \ker \phi$$r-r'\in\ker\varphi and hence $$r+\ker \phi =r^{\prime }+\ker \phi$$r+\ker\varphi=r'+\ker\varphi. Thus $$\mathrm{\Psi }$$\Psi is injective, and

Both proofs express the same structure: after collapsing the kernel and then mapping, exactly the image remains.

6Exercise link

7Summary

The first isomorphism同型どうけい theorem says that "quotienting by the kernel核かく gives the image像ぞう." The same form appears for groups, rings, and linear maps, making it an important overview that connects abstract algebra抽象代数ちゅうしょうだいすう with linear algebra.