Group Homomorphisms and Isomorphisms

At the heart of abstract algebra is the desire to compare algebraic structures. When are two groups essentially the same? How can we map one group into another while respecting its fundamental operation? These questions are answered by the study of homomorphisms (structure-preserving maps) and isomorphisms (perfect structure-preserving correspondences). Mastering these concepts and the powerful isomorphism theorems that follow is crucial for analyzing group structure, simplifying complex groups via quotients, and establishing profound correspondences between subgroups.

Homomorphisms and the Preservation of Structure

A group homomorphism is a function $\varphi :G\to H$ between two groups $(G,\cdot )$ and $(H,\ast )$ that preserves the group operation. Formally, for all $a,b\in G$, we have $\varphi (a\cdot b)=\varphi (a)\ast \varphi (b)$. This simple condition has immediate, important consequences. It forces the homomorphism to map the identity of $G$ to the identity of $H$: $\varphi (e_G)=e_H$. Furthermore, it preserves inverses: $\varphi (a^{-1})=(\varphi (a){)}^{-1}$ for all $a\in G$.

Consider the map $\varphi :\mathbb{Z}\to {\mathbb{Z}}_n$ defined by $\varphi (k)=k\mathrm{mod}n$. This is a homomorphism from the additive group of integers to the additive group of integers modulo $n$, because the remainder of a sum is the sum of the remainders: $\varphi (a+b)=(a+b)\mathrm{mod}n=(a\mathrm{mod}n){+}_n(b\mathrm{mod}n)=\varphi (a){+}_n\varphi (b)$. This example is foundational and leads directly to quotient group construction.

The most stringent type of homomorphism is an isomorphism. An isomorphism is a homomorphism that is both injective (one-to-one) and surjective (onto). If an isomorphism exists between groups $G$ and $H$, we write $G\cong H$ and say the groups are isomorphic. Isomorphic groups are structurally identical; they may differ in the labels of their elements but share identical group-theoretic properties like order, the pattern of subgroups, and being abelian or cyclic.

Kernels and Images: Natural Subgroups

Every homomorphism $\varphi :G\to H$ gives rise to two intrinsically important subgroups. The image of $\varphi$, denoted $\text{Im}(\varphi )$ or $\varphi (G)$, is the set $\{\varphi (g):g\in G\}$ in $H$. It is always a subgroup of $H$. The kernel of $\varphi$, denoted $\text{Ker}(\varphi )$, is the set of elements in $G$ that map to the identity in $H$: $\{g\in G:\varphi (g)=e_H\}$. The kernel is more than just a subgroup; it is a normal subgroup of $G$.

A subgroup $N$ of $G$ is normal (written $N⊴G$) if $gN{g}^{-1}=N$ for every $g\in G$. Equivalently, $gN=Ng$ for all $g$, meaning left and right cosets coincide. The kernel of any homomorphism is always normal. Conversely, and profoundly, if $N⊴G$, then $N$ is the kernel of the natural canonical homomorphism $\pi :G\to G/N$ defined by $\pi (g)=gN$. This map, which sends each element to its coset, is surjective, and its kernel is exactly $N$. This establishes a critical duality: normal subgroups are precisely the kernels of homomorphisms.

The First Isomorphism Theorem

The First Isomorphism Theorem is the cornerstone that connects homomorphisms, kernels, images, and quotient groups. It states: Let $\varphi :G\to H$ be a group homomorphism. Then the kernel $K=\text{Ker}(\varphi )$ is a normal subgroup of $G$, and we have an isomorphism $$G/K\cong \text{Im}(\varphi ).$$ The isomorphism is given explicitly by the map $\psi :G/K\to \text{Im}(\varphi )$ defined by $\psi (gK)=\varphi (g)$. You must verify this map is well-defined (i.e., if $gK=hK$, then $\varphi (g)=\varphi (h)$), a homomorphism, injective, and surjective onto the image.

This theorem is incredibly powerful. It tells us that the image of any homomorphism is isomorphic to a quotient of the domain by the kernel. It also shows that homomorphisms essentially "factor through" their kernels. For our earlier example $\varphi :\mathbb{Z}\to {\mathbb{Z}}_n$, the kernel is the set of multiples of $n$, which is the subgroup $n\mathbb{Z}$. The theorem confirms the fundamental isomorphism $\mathbb{Z}/n\mathbb{Z}\cong {\mathbb{Z}}_n$.

The Second and Third Isomorphism Theorems

The Second and Third Isomorphism Theorems are powerful tools for manipulating quotient groups and understanding the lattice of subgroups.

The Second Isomorphism Theorem (The Diamond Isomorphism Theorem): Let $G$ be a group, $A$ a subgroup of $G$, and $N$ a normal subgroup of $G$. Then:

1. The product set $AN=\{an:a\in A,n\in N\}$ is a subgroup of $G$.
2. $N$ is a normal subgroup of $AN$.
3. $A\cap N$ is a normal subgroup of $A$.
4. We have the isomorphism:

$$A/(A\cap N)\cong AN/N.$$ This theorem is often visualized with a subgroup lattice diagram shaped like a diamond. It allows you to "cancel" $N$ in a quotient when $N$ is part of a larger subgroup containing $A$.

The Third Isomorphism Theorem (The Freshman Theorem): Let $G$ be a group with normal subgroups $N$ and $K$ such that $N\subseteq K\subseteq G$. Then $K/N$ is a normal subgroup of $G/N$, and we have the isomorphism: $$(G/N)/(K/N)\cong G/K.$$ This theorem states that taking successive quotients is equivalent to taking a single quotient by the larger normal subgroup. It's analogous to simplifying fractions: $(G/N)/(K/N)\cong G/K$.

Applications to Subgroup Structure and Correspondence

The theorems, particularly the First and the Correspondence Theorem (often considered a Fourth Isomorphism Theorem), are indispensable for analyzing group structure. The Correspondence Theorem states: Let $G$ be a group and $N⊴G$. Then there is a bijective, inclusion-preserving correspondence between the subgroups of $G$ that contain $N$ and the subgroups of the quotient group $G/N$.

Furthermore, under this correspondence:

• Normal subgroups of $G$ containing $N$ correspond to normal subgroups of $G/N$.
• If $H$ is a subgroup of $G$ containing $N$, then $|G:H|=|(G/N):(H/N)|$.

This is a profound organizational tool. When analyzing a complex group $G$, you can often find a suitable normal subgroup $N$, study the simpler quotient group $G/N$, and then lift all your findings about the subgroups of $G/N$ back to a complete understanding of the subgroups of $G$ that lie "above" $N$. This technique is fundamental in classifying groups of a given order, analyzing solvable groups, and studying group extensions.

Common Pitfalls

1. Assuming all homomorphisms are isomorphisms. A homomorphism need not be injective or surjective. Confusing a general homomorphism with an isomorphism leads to incorrect conclusions about the structure of the domain or codomain. Always check the properties of the kernel and image separately.
2. Misapplying the isomorphism theorems without verifying normality. The Second Theorem requires $N$ to be normal in $G$, not just in $A$. The Third Theorem requires a chain of normal subgroups $N⊴K⊴G$. Applying these theorems when the normality conditions are not met is a critical error.
3. Confusing the group elements with their cosets in quotients. In a quotient group $G/N$, the elements are sets (cosets). A statement like "$g=e$ in $G/N$" is false; the correct statement is "$gN=N$" or "$g\in N$". This distinction is vital for correctly applying the canonical homomorphism and the isomorphism theorems.
4. Forgetting that the Correspondence Theorem is for subgroups containing the kernel. When using the First Isomorphism Theorem, the correspondence is between subgroups of $G/K$ and subgroups of $G$ that contain $K$. It is not a correspondence with all subgroups of $G$.

Summary

• A group homomorphism $\varphi :G\to H$ preserves the group operation ($\varphi (ab)=\varphi (a)\varphi (b)$). Its image is a subgroup of $H$, and its kernel is a normal subgroup of $G$.
• An isomorphism is a bijective homomorphism, indicating two groups have identical structure ($G\cong H$).
• The First Isomorphism Theorem ($G/\text{Ker}(\varphi )\cong \text{Im}(\varphi )$) is the fundamental link between homomorphisms and quotient groups.
• The Second Isomorphism Theorem allows you to relate intersections and products of subgroups in a quotient: $A/(A\cap N)\cong AN/N$.
• The Third Isomorphism Theorem simplifies iterated quotients: $(G/N)/(K/N)\cong G/K$ for $N\subseteq K⊴G$.
• The Correspondence Theorem provides a powerful lattice isomorphism between subgroups of $G/N$ and subgroups of $G$ containing $N$, enabling structured analysis of complex groups.