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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 \mathbb{P}^1.  Within this broad framework, two classes are of particular interest:

1. The group of biholomorphic maps f: \mathbb{P}^1 \to \mathbb{P}^1 (those that respect the structure of \mathbb{P}^1 as a Riemann surface). It is well-known that such maps are given by Möbius transformations, i.e. rational functions of the form

\displaystyle f(z) = \frac{a z +b}{cz +d},

satisfying ad-bc \neq 0 \in \mathbb{C}. The group of Möbius transformations (also known as the Möbius Group and herein denoted \mathcal{M}) is naturally isomorphic to \mathrm{PSL}_2(\mathbb{C}), the projective (special) linear group, via:

\displaystyle\frac{az+b}{cz+d} \mapsto \left(\begin{matrix} a & b \\ c & d \end{matrix} \right).

2. The group of conformal maps f: \mathbb{P}^1 \to \mathbb{P}^1, denoted \mathcal{C} for brevity.  To be clear, here we refer to those maps f 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 \mathbb{P}^1 is either biholomorphic (returning to case (1)), or bijective and anti-holomorphic: a biholomorphic function of the complex conjugate \overline{z}.

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

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