Proof of the First Theorem of Group Isomorphism

The First Theorem of Group Isomorphism, also known as the First Isomorphism Theorem, provides a deep connection between the structure of a group and its subgroup. Specifically, it states that if we have a surjective homomorphism from a group G to a group H, then the quotient of G by the kernel of the homomorphism is isomorphic to H. In this article, we will prove this theorem step by step.

Setting the Stage: Homomorphisms and Surjective Functions

Let us consider a surjective homomorphism φ: G → H. Our goal is to prove that G/K ? H, where K is the kernel of φ. The kernel of φ, denoted as K, is defined as the set of all elements in G that map to the identity element of H. That is, K {g ∈ G | φ(g) e_H}.

The Proof: Establishing Key Properties

To prove that G/K ? H, we need to establish the following properties of the function f: G/K → H defined by f(gK) φ(g), where gK represents the coset of K containing g in G/K.

1. Well-Definedness

The first property we need to prove is that f is well-defined. In other words, if two elements in G/K represent the same coset, then they must map to the same element in H.

Proof: Suppose g_1K g_2K. This implies that g_1 and g_2 are in the same coset of K. Therefore, there exists an element z in K such that g_1 g_2z. We then show that f(g_1K) f(g_2K): begin{align*}f(g_1K) f(g_2zK) φ(g_2z) φ(g_2)φ(z) φ(g_2)φ(e_H) φ(g_2) f(g_2K).end{align*}

2. Homomorphism

Next, we need to prove that f is a homomorphism. This means that for any two cosets g_1K and g_2K in G/K, the following holds:

Proof: Let g_1K, g_2K be any two cosets in G/K. Then: begin{align*}f(g_1Kg_2K) f((g_1g_2)K) φ(g_1g_2) φ(g_1)φ(g_2) f(g_1K)f(g_2K).end{align*}

3. Injectivity

To show that f is injective, we need to prove that its kernel is trivial. In other words, if f maps an element to the identity of H, then that element must be the identity of G/K.

Proof: Suppose f(gK) e_H. This implies that φ(g) e_H. Since φ(K) {e_H}, it follows that g is an element of K. Therefore, gK e_GK e_{G/K}, which means that f has a trivial kernel and is injective.

4. Surjectivity

Finally, we need to prove that f is surjective. This means that every element in H is the image of some coset in G/K.

Proof: Since φ is a surjective homomorphism, for every element h in H, there exists an element g in G such that φ(g) h. Then, f(gK) φ(g) h, which shows that f is surjective.

Conclusion

By proving the well-definedness, homomorphism, injectivity, and surjectivity of f, we have established that G/K ? H. This completes the proof of the First Theorem of Group Isomorphism.

Understanding and mastering the First Isomorphism Theorem is crucial for delving deeper into group theory and its applications in various fields of mathematics and beyond. Whether you are a student of abstract algebra, a mathematician, or a researcher in related areas, this theorem provides a powerful tool for understanding the structure of groups and homomorphisms.