Intuition behind the the definition of homomorphisms and isomorphisms

Question

I'm familiar with the definition (inverses and bijections, preserving operations) in the context of groups and vector spaces and do know the formal definitions of morphisms as a whole.

However, what I'm looking for is a mathematical justification and a sharp intuition for the purpose of defining homomorphisms and isomorphisms. I find it quite challenging to picture them and find myself unable to convince myself when I'm trying to explain those concepts.

Right now, the only analogy I can make is one for the isomorphisms in the context of groups, namely I like to think about the isomorphism between $\mathbb{Z}_2$ with the addition and the set {$-1,1 $ } with multiplication where we map $0$ to $1$ and $1$ to $-1$, this way adding two even numbers or two odd numbers is analogous to multiplying elements of {$-1,1$ } , although when I try to abstract this example further I hit some brick walls :

• Making Sense of $f(a) f(b)=f(a \star b)$ where juxtaposition is the operation of a group $G$ and the star is the operation of a group $H$
• I cannot see how the structure of the group $G$ is being preserved in $H$.

I would like an answer giving intuition about those concepts rather than their formal definitions. Note that I already consulted other threads concerning this question, so this post aims to look at other perspectives in hope to understand better.

As i have already poined the out , i've already seen and read the linked thread multiple times but did not personally get a satisfactory answer , what i'm asking for is to shed some light upon how the preserving of group structure is being translated through $f(a) f(b)=f(a \star b)$ .

Answer 1

The basic intuition behind homomorphisms is the following: if $G$ and $H$ are two algebraic structures of the same type (e.g. groups, vector spaces, etc.), and $\varphi:G\to H$ is a homomorphism, then a homomorphism is a "structure-preserving" map. Now, what does "structure-preserving" mean? This term is a little loose, and it easiest to understand it with examples:

• Every group has a binary operation, an identity element, and inverses for every element. These three "operations" comprise the structure of the group. If $\varphi:G\to H$ is a group homomorphism, then $\varphi$ "preserves" the structure in the sense that $\varphi(a)\varphi(b)=\varphi(ab)$, $\varphi(e_G)=e_H$, and $\varphi(a^{-1})=\varphi(a)^{-1}$. It turns out that simply requiring the first condition ensures that the other two hold, hence the definition given in textbooks. These three equations have a common thread to them which I will try to distill here. For each $a\in G$, it helps to think of $\varphi(a)$ as the "corresponding element" to $a$: the way in which $a$ interacts with other elements of $G$ is similar to the way $\varphi(a)$ interacts with other elements of $H$. For instance, if an equation $ab=c$ holds in $G$, then replacing $a$, $b$, and $c$ with each of their corresponding elements gives us the equation $\varphi(a)\varphi(b)=\varphi(c)$, which holds in $H$. Similarly, if $a^{-1}=b$, then $\varphi(a)^{-1}=\varphi(b)$. Finally, a group homomorphism sends the identity of $G$ to the identity of $H$.
• The situation with vector spaces is similar, except that there is more structure to be preserved than with groups. A vector space homomorphism, which for historical reasons tends to be called a "linear map", preserves all the operations that a group homomorphism does, but it also preserves scalar multiplication. It turns out that simply requiring that $\varphi(u+v)=\varphi(u)+\varphi(v)$ and $\varphi(\lambda v)=\lambda\varphi(v)$ suffices. (Warning: in a vector space, the group operation is written using additive notation rather than multiplicative notation, and scalar multiplication is written using multiplicative notation; don't let this confuse you.)

An isomorphism between $G$ and $H$ is an invertible homomorphism: that is, it is a homomorphism from $G$ to $H$ which has an inverse that it also a homomorphism. In the case of algebraic structures such as groups and vector spaces, every bijective homomorphism is an isomorphism. (In more exotic categories, such as the category of topological spaces, this is not so: a bijective continuous map does not necessarily have a continuous inverse.)

If there exists an isomorphism between $G$ and $H$, then they are said to be isomorphic. Isomorphic structures are often regarded as "the same", but this is a more subtle concept than you might think. In Michael Artin's Algebra, he gives the following analogy:

When $\varphi:G\to G'$ is an isomorphism, we can make a computation in either group. then use $\varphi$ or $\varphi^{-1}$ to carry it over to the other. So, for computation with the group law, the two groups have identical properties. To picture this conclusion intuitively, suppose that the elements of one of the groups are put into unlabeled boxes, and that we have an oracle that tells us, when presented with two boxes, which box contains their product. We will have no way to decide whether the elements in the boxes are from $G$ or from $G'$.

It is instructive to think hard about why the formal definition really does align with the "box analogy". Let me know if you have any questions.