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In this post, we discuss a few ways in which the symmetric and alternating groups can be realized as finite collections of self-maps on the Riemann sphere. Throughout, our group operation will be composition of functions: as such, the maps we choose will necessarily be homeomorphisms of . Within this broad framework, two classes are of particular interest:

1. The group of biholomorphic maps (those that respect the structure of as a Riemann surface). It is well-known that such maps are given by Möbius transformations, i.e. rational functions of the form

satisfying . The group of Möbius transformations (also known as the *Möbius Group* and herein denoted ) is naturally isomorphic to , the projective (special) linear group, via:

2. The group of *conformal* maps , denoted for brevity. To be clear, here we refer to those maps which preserve **unsigned** angle measure. *(In contrast, some authors require conformal maps to preserve orientation as well.)* We recall the fundamental result that such maps contain the Möbius group as a subgroup of index two. To be specific, any conformal self-map on is either biholomorphic (returning to case (1)), or bijective and *anti-holomorphic*: a biholomorphic function of the complex conjugate .

After the fold, we begin a two-part program to calculate the maximal such that the symmetric group injects into (resp. ). Along the way, we study injections of the alternating group into , and highlight some exceptional cases in which our injections can be attached to group actions on a finite invariant set.

In differential calculus, the product rule is both simple in form and high in utility. As such, it is typically presented early on in calculus courses — soon after the linearity of the derivative, in fact. Moreover, the product rule is easy to derive from first principles:

**Theorem (Product Rule): ***Let and be differentiable on the open set . Then is differentiable on , and we have*

for all *.*

*Proof: *For , we have (by definition of the derivative)

under the assumption that each of these last two limits exists. This of course holds, as these limits are and , respectively.

All in all, then, the product rule is easy to prove and easy to use. But — and this is of utmost pedagogical importance — * is the product rule intuitive? *By this proof alone, I would argue not; the manipulation of the numerator is weakly-motivated and our result falls out without reference to more general phenomena.

In this post, we’ll explore the merits of a second proof of the product rule, one that I hope presents a motivated and compelling argument as to **why**** **the product rule should look the way it does.