Applications of the concept of homomorphism What are some interesting applications of the concept of homomorphism?
Example: If there is a homorphism from a ring $R$ to a ring $r$ then a solution to a polynomial equation in $R$ gives rise to a solution in $r$. e.g. if $f:R \rightarrow r$ and $X^2+Y^2=0$ then $f(X^2+Y^2)=f(0), f(X^2)+f(Y^2)=0, f(X)^2+f(Y)^2=0, x^2+y^2=0$

Here are some striking examples from looking at the Lebesgue space $L^1(\mathbb{R})$ under Fourier transform. $L^1(\mathbb{R})$ is an algebra under convolution, that is it is a vector space with a multiplication (here multiplication is the convolution). The Fourier transform $\mathcal{F}: L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an algebra homomorphism from $L^1$ into a sub-algebra $\mathcal(L^1(\mathbb{R}))$ of the space $C_0$ (continuous functions that vanish at $\pm\infty$). 


*

*In particular 
$$\mathcal{F}(f*g) = \mathcal{F}(f)\cdot\mathcal{F}(g)$$ 
and instead of studying convolution equations in $L^1$ we may study multiplicative equations in the Fourier image. 

*An other example is The Wiener Tauberian Theorem, that states that the translates of $f$ span a dense subspace of $L^1$ if and only if $\mathcal{F}(f)$ is non-zero.
