Consider the simple function of multiplying a number times itself. I am interested in knowing more about the kind of groups which may be devised over which this function is an injective homomorphism. The most trivial among these, I would say, is where we define the function $x^2:(\mathbb{R}^{+}/\{0\},\times) \rightarrow (\mathbb{R}^{+}/\{0\},\times)$, where $\times$ of course is elementary multiplication. This corresponds to cutting-off half the graph like was done in school for $x=y^2$. But what are some more exotic groups?
Requirements
- $|\ker x^2| = 1$
- Operation preservation
- Domain, codomain should be groups (or not, go crazy)
- Definition must roughly correspond to squaring a number in some meaningful way (this is intentionally open-ended)
- Challenge: no graphs
I am currently reading about the theorems of homomorphisms and I wanted to gain more insight into the sort of groups/structures I should expect to see going into my future courses. This is not a homework problem. I will also post here anything I find for fun though I totally expect to be outclassed. Thank you.