# Where can I look to investigate and learn about broader classes of symmetries for algebraic structures?

Here's $$f(x) = x^2,$$ it has the property that $$f(-x) = f(x).$$

I have a couple questions related to this topic:

1.) Is there a broader name for symmetries of the form $$f(h(x)) = f(x)$$ for invertible $$h$$?

2.) Is there a broader name of symmetries that keep a manifold invariant, for the cases that a manifold is defined by a system of equations like $$x^2 + y = z$$, $$z^2 - 1 = y^2$$ and so on?

3.) How do we keep the "symmetry" invariant of the specific coordinates? Every parabola has a line of symmetry about its vertex, but in the example I provided, the line of symmetry is stuck at the line $$x = 0,$$ how do we classify and find symmetries that are irregardless of the specific coordinates, to say that a parabola has a symmetry about some line centered at its vertex?

4.) How do we know we've found all symmetries of a given manifold, at least in 2D or 3D cases?

• Okay, let's take one question one at a time then. I'm interested in symmetries of the form $f(h(x)) = f(x),$ the generalization of $f(-x) = f(x).$ What branch of math most closely studies this? $f(-x) = f(x)$ is an example of a discrete symmetry, but could it be a holomorphic symmetry if you interpolate over complex numbers? Oct 10, 2022 at 18:03
• @askquestions4: yes, one could ask for continuous families of symmetries, or smooth or even holomorphic. This is still a very general question so many branches of math are relevant, as I mentioned in my answer to your other question: group theory, Lie theory, representation theory, harmonic analysis... it would help if you got even more specific than this, e.g. naming a specific function $f$ and a specific class of symmetries $h$ you want to characterize (algebraic, continuous, smooth, holomorphic, etc.). Oct 10, 2022 at 18:17
• But what I'm interested in is as general as I presented it. I guess we could be more specific to start with $f$ and $h$ being analytic. Given $f$, what is the name or process by which we find symmetries such that $f(h(x)) = f(x)$? Oct 10, 2022 at 18:19
• @askquestions4: it's just called "finding the symmetries," I guess. Again, you are asking an extremely general question. I am going to specifically limit attention to the function $f(x) = x^2$ on $\mathbb{R}$ for simplicity. A basic observation is that any function $h$ satisfying $f(h(x)) = f(x)$ must permute the level sets $f^{-1}(r) = \{ \pm \sqrt{r} \}$. Equivalently, for any particular $x$ we must have $h(x) = x$ or $h(x) = -x$. It follows that if $h$ is continuous and invertible then the only possibilities are $h(x) = x$ for all $x$ or $h(x) = -x$ for all $x$. Oct 10, 2022 at 18:30
• This example was pretty easy to deal with because the level sets were so small. If $f$ had been a function $\mathbb{R}^n \to \mathbb{R}$ then its level sets would be manifolds (or more complicated objects) and things would get more complicated. Oct 10, 2022 at 18:31