maths.free › Abstract Algebra › Homomorphisms › The Isomorphism Theorems

The Isomorphism Theorems

Although it is not evident at first, factor groups correspond exactly to homomorphic images, and we can use factor groups to study homomorphisms.

The Isomorphism Theorems

Although it is not evident at first, factor groups correspond exactly to homomorphic images, and we can use factor groups to study homomorphisms. We already know that with every group homomorphism \(\phi: G \rightarrow H\) we can associate a normal subgroup of \(G\), \(\ker \phi\). The converse is also true; that is, every normal subgroup of a group \(G\) gives rise to homomorphism of groups.

Let \(H\) be a normal subgroup of \(G\). Define the natural or canonical homomorphism \[\begin{aligned}\end{aligned}\] by \[\begin{aligned}\end{aligned}\]. This is indeed a homomorphism, since \[\begin{aligned}\end{aligned}\]. The kernel of this homomorphism is \(H\). The following theorems describe the relationships between group homomorphisms, normal subgroups, and factor groups.

Theorem: First Isomorphism Theorem

If \(\psi : G \rightarrow H\) is a group homomorphism with \(K =\ker \psi\), then \(K\) is normal in \(G\). Let \(\phi: G \rightarrow G/K\) be the canonical homomorphism. Then there exists a unique isomorphism \(\eta: G/K \rightarrow \psi(G)\) such that \(\psi = \eta \phi\).

Proof

We already know that \(K\) is normal in \(G\). Define \(\eta: G/K \rightarrow \psi(G)\) by \(\eta(gK) = \psi(g)\). We first show that \(\eta\) is a well-defined map. If \(g_1 K =g_2 K\), then for some \(k \in K\), \(g_1 k=g_2\); consequently, \[\begin{aligned}\end{aligned}\]. Thus, \(\eta\) does not depend on the choice of coset representatives and the map \(\eta: G/K \rightarrow \psi(G)\) is uniquely defined since \(\psi = \eta \phi\). We must also show that \(\eta\) is a homomorphism. Indeed, \[\begin{aligned}\eta( g_1K g_2K ) & = \eta(g_1 g_2K) \\ & = \psi(g_1 g_2) \\ & = \psi(g_1) \psi(g_2) \\ & = \eta( g_1K) \eta( g_2K )\end{aligned}\]. Clearly, \(\eta\) is onto \(\psi( G)\). To show that \(\eta\) is one-to-one, suppose that \(\eta(g_1 K) = \eta(g_2 K)\). Then \(\psi(g_1) = \psi(g_2)\). This implies that \(\psi( g_1^{-1} g_2 ) = e\), or \(g_1^{-1} g_2\) is in the kernel of \(\psi\); hence, \(g_1^{-1} g_2K = K\); that is, \(g_1K =g_2K\).

Mathematicians often use diagrams called commutative diagrams to describe such theorems. The following diagram commutes since \(\psi = \eta \phi\).

Notice that in the course of the proof of , we have also proved the following theorem.

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.

今度はあなたの番計算機では解けませんが、計算可能です。下の一つを試してみましょう。もしくは自分で入力してください。

自分の解答を記録しよう

無料アカウントでは、すべてのレッスンにノートを追加し、完成したことの記録、解いた問題を一つの場所に保存し、このページについて質問できる先生を追加します。数学自体はログインしたかどうかに関係なく誰でも利用できます。

登録ログイン

ここで使用された記号

どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。

よくある質問

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). ここでは要約して説明し直しています。誤りは私たちの責任です。

More in Abstract Algebra

Groups Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula