What's a name for a general technique I've seen used many times?


Given any family $\mathcal{F}$ of functions such that $f:X\to Y$ for all $f\in \mathcal{F}$ when one wishes to study in general for an arbitrary $y\in Y$ what (if any) $x\in X$ satisfy $f(x)=y$ it seems often the approach taken is to first find some permutation group $P\subseteq \text{Sym}(Y)$ such that:


$$\forall \sigma\in P\left(f\in \mathcal{F}\implies f\circ \sigma\in \mathcal{F}\right)$$


Then from here viewing this as a right group action of $P$ on $\mathcal{F}$ we form an equivalence relation $\sim$ over $\mathcal{F}$ with each equivalence class an orbit e.g. $\text{Orb}(f)=\{f\circ \sigma\in \mathcal{F}:\sigma\in P\}$ so that we get:


$$f\sim g\iff \text{Orb}(f)=\text{Orb}(g)\iff \exists \sigma\in P:f=g\circ \sigma$$


At this point for every $f\in \mathcal{F}$ given the equation $f(x)=y$ we find a unique "nice" $C\_{f}\in \text{Orb}(f)$ (nice in its easier to study what $x\in X$ satisfy $C\_f(x)=y$) and let this be a canonical form for our equivalence class $\text{Orb}(f)=[f]\_{\sim}$ then our original problem reduces to just studying the family of functions $\mathcal{G}\subseteq \mathcal{F}$ where $\mathcal{G}=\{C\_f\in \mathcal{F}:f\in \mathcal{F}\}$. To make this idea more explict here are three examples where I've seen this sort of idea used:




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A simple first example is the notion of "reduced echelon form" in introductory linear algebra courses where here $\mathcal{F}=\mathcal{M}\_{m\times n}(\mathbb{F})$ would be the set of all matrices over some field $\mathbb{F}$ and $P=\text{GL}\_n(\mathbb{F})$ would be a general linear group over $\mathbb{F}$. <https://en.wikipedia.org/wiki/Row_echelon_form>




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A second example is many techniques used when studying univariate polynomials over again just to make things simple lets say those polynomials with coefficients in $\mathbb{F}$ so we get $\mathcal{F}\subseteq\mathbb{F}[X]$. From here $P$ would take the form of scaling/shifting for example reducing each polynomial to a monic one, or shifting along its input to eliminate a monomial in the resulting composition. E.g. if we have:


$$f(x)=x^n+ax^{n-1}+bx^{n-2}+\cdots+c$$


Then if $\sigma:x\to x-\frac{a}{n}$ expanding $(f\circ \sigma)(x)$ the coefficient on the monomial $x^{n-1}$ vanishes. Where in particular if $1<n<5$ this is specifically named/detailed out here:


When $n=2$: <https://en.wikipedia.org/wiki/Completing_the_square#General_description>


When $n=3$: <https://en.wikipedia.org/wiki/Cubic_function#Reduction_to_a_depressed_cubic>


When $n=4$: <https://en.wikipedia.org/wiki/Quartic_function#Converting_to_a_depressed_quartic>




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A third example comes when studying positive definite integral binary quadratic forms so that here $\mathcal{F}$ is the set of BQFs and $P=\text{GL}\_2(\mathbb{Z})$ thus reducing each $f\in \mathcal{F}$ with $f(x,y)=ax^2+bx+c$ through a linear transformation of its input $(x,y)$ via multiplication by matrices in $\text{GL}\_2(\mathbb{Z})$ until we have a BQF satisfying $|2|c|-\sqrt{b^2-4ac}|<b<\sqrt{b^2-4ac}$ is how we get our canonicals also for negative discriminants see: <http://mathworld.wolfram.com/ReducedBinaryQuadraticForm.html>




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So to re-iterate my question, does this general "technique" I outlaid at the start have a name? Or is it just seen as an obvious approach/tool to try and use when studying a given family of functions?



Classifying functions up to suitable pre-composition and/or post-composition
The emphasis on finding solutions to equations $f(x) = y$ is a red herring, and note that in your examples your symmetries aren't acting pointwise on $Y$ as in your general explanation (sometimes they are acting on $X$), but that's not particularly important. 


More generally, suppose you want to understand anything about the elements of any set $S$ whatsoever; a very common approach to doing this is to find a group $G$ acting on $S$ such that the thing you want to understand is invariant under the action of $G$, then find a nice normal or canonical form under the action of $G$. 


Terence Tao calls this **[spending symmetry](https://terrytao.files.wordpress.com/2012/11/blog-book.pdf)**; related notions include


* [fixing](https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/) a [gauge](https://en.wikipedia.org/wiki/Gauge_fixing),
* finding a [fundamental domain](https://en.wikipedia.org/wiki/Fundamental_domain) for a group action.


Here are two quick examples more general than solving equations:


* The spectral theorem says that every orbit under the conjugation action of the unitary group on normal matrices contains a diagonal matrix. This makes it easy, among other things, to exponentiate normal matrices, and more generally can be used to establish functional calculus for matrices.
* The existence of singular value decomposition says that every orbit under the action of $O(n) \times O(m)$ on $n \times m$ real matrices (by left and right multiplication) contains a diagonal matrix with nonnegative entries. This has many applications, for example to describing low rank approximations to matrices; see e.g. [this blog post](https://qchu.wordpress.com/2017/03/13/singular-value-decomposition/).


Groups naturally appear as changes of coordinates in various senses, so there are lots of examples involving groups, but there are also important examples where something more general than a group action appears; the more general thing might be called **[reductions](https://en.wikipedia.org/wiki/Reduction_(complexity))**. 


Again suppose you are trying to understand anything about the elements of any set $S$ whatsoever. Moreover, suppose you can establish "implications" between the elements of $S$, to the effect that if you've understood some $s\_1 \in S$, then you've understood some $s\_2 \in S$; we can capture these implications in a relation $R$ on $S$ (which, in the group action example, is the relation of belonging to the same orbit). Under the very mild assumption that this relation $R$ is reflexive and transitive, $R$ gives $S$ the structure of a [preorder](https://ncatlab.org/nlab/show/preorder) (which in the group action case is an equivalence relation), and to understand all of $S$ it suffices to understand some subset $S\_0 \subseteq S$ of elements which "cover" $S$ in the sense that implications from $S\_0$ reach all of $S$. 


Maybe the most famous example of this phenomenon is in computer science, where if $S$ is the set of problems in NP, then $S\_0$ can be taken to be the set with a single element consisting of any [NP-complete problem](https://en.wikipedia.org/wiki/NP-completeness). Here the relation $R$ is [polynomial-time reduction](https://en.wikipedia.org/wiki/Polynomial-time_reduction). But mathematicians also use reductions all the time, any time they end up saying something like "...so the problem reduces to understanding the special case where...." A typical mathematical example is using techniques like [localization](https://en.wikipedia.org/wiki/Localization_of_a_ring) in commutative algebra to reduce a problem about arbitrary commutative rings to a particularly simple class of rings, maybe noetherian rings, local rings, or fields. 


