Applications of the concept of homomorphism What are some

When you are studying a particular structure, you can study it in isolation: you can consider a vector space, and its subsets and subspaces, and say a lot of interesting things about that vector space; but in the end, it doesn't get you all that far. The same is true when you study a particular ring (and you can learn a lot just by staring at that particular ring; think of classical Number Theory as the result of staring intently at the integers looking for interesting features), or groups, or topological spaces, etc. But it turns out that studying the functions between similar objects that preserve the structure (linear transformations for vector spaces, ring and group homomorphisms, continuous maps, etc) leads to a much richer palette, with a lot more information and a lot more structure to work with. In some cases, like vector spaces, the new objects have themselves the same structure that you were studying (the set of all linear transformations ${\displaystyle T:V\to W}$ can be given a natural structure of a vector space, so everything you know about vector spaces applies) or some other structure (if A and B are abelian groups, then the set of all group homomorphism ${\displaystyle A\to B}$ can be given a ring structure by pointwise addition and composition, so you can study it as you would study a ring) which in turn gives you a wealth of information about the structures themselves. And of course, things like the Isomorphism Theorems can give you a lot of information about the structure you are looking at by looking at its images, which will often be "simpler" or "smaller" than the one you were originally concerned with.
Even in the case were we have found a lot of good information by simply staring (as in classical number theory), just look at how big a leap forward was achieved with the introduction of congruences (homomorphisms from ${\displaystyle \mathbb{Z}}$ to ${\displaystyle \frac{\mathbb{Z}}{m}\mathbb{Z}}$), or with algebraic number theory (mapping to larger rings and then trying to import the information back into ${\displaystyle \mathbb{Z}}$).

Here are some striking examples from looking at the Lebesgue space ${\displaystyle L^1\left(\mathbb{R}\right)}$ under Fourier transform. ${\displaystyle L^1\left(\mathbb{R}\right)}$ is an algebra under convolution, that is it is a vector space with a multiplication (here multiplication is the convolution). The Fourier transform
${\displaystyle \mathcal{F}:L^1\left(\mathbb{R}\right)\to C_0\left(\mathbb{R}\right)}$ is an algebra homomorphism from ${\displaystyle L^1}$ into a sub-algebra ${\displaystyle \left(L^1\left(\mathbb{R}\right)\right)}$ of the space ${\displaystyle C_0}$ (continuous functions that vanish at ${\displaystyle \pm \mathrm{\infty }}$).
1. In particular ${\displaystyle \mathcal{F}(f\cdot g)=\mathcal{F}\left(f\right)\cdot \mathcal{F}\left(g\right)}$
and instead of studying convolution equations in ${\displaystyle L^1}$ we may study multiplicative equations in the Fourier image.
2. An other example is The Wiener Tauberian Theorem, that states that the translates of f span a dense subspace of ${\displaystyle L^1}$ if and only if ${\displaystyle \mathcal{F}\left(f\right)}$ is non-zero.