What is known about this group reminiscent of the anharmonic group?

Question

The anharmonic group is this nonabelian group of six rational functions with the operation of composition of functions: \begin{align} t & \mapsto t & & \text{order 1} \\[8pt] t & \mapsto 1/t & & \text{order 2} \\ t & \mapsto 1-t & & \text{order 2} \\ t & \mapsto t/(t-1) & & \text{order 2} \\[8pt] t & \mapsto 1/(1-t) & & \text{order 3} \\ t & \mapsto (t-1)/t & & \text{order 3} \end{align} The reason it is called "anharmonic" appears to be that a set of four numbers is said to divide the line harmonically if their cross-ratio is $1$, and so the cross-ratio measures deviation from harmonic division, and when four numbers with cross-ratio $t$ are permuted, this group gives the six values that the cross-ratio can take. The members of this group permute the elements $0$, $1$, and $\infty$ of $\mathbb C\cup\{\infty\}$.

Today I noticed that something very similar-looking forms a group of four elements, each of the three non-identity elements having order $2$: \begin{align} t & \mapsto t \\ t & \mapsto -1/t \\ t & \mapsto (1-t)/(1+t) \\ t & \mapsto (t+1)/(t-1) \end{align}

• Can anything interesting be said about this, including, but not limited to, relevance to geometry, algebra, combinatorics, probability, number theory, physics, or engineering?
• What other finite groups of rational functions as simple as these exist?

Answer 1

The group of invertible rational functions, which explicitly consists of Mobius transformations $z \mapsto \frac{az + b}{cz + d}$, is abstractly the group $PSL_2(\mathbb{C})$. Its finite subgroups are known: by an averaging argument they correspond to finite subgroups of $PSU(2) \cong SO(3)$, the group of orientation-preserving isometries of the sphere $S^2$. There are two infinite sequences of such subgroups, the cyclic groups $C_n$ and the dihedral groups $D_n$, and then three "exceptional" groups given by the symmetry groups of the Platonic solids:

• $A_4$, the symmetry group of the tetrahedron.
• $S_4$, the symmetry group of the cube and the octahedron.
• $A_5$, the symmetry group of the icosahedron and dodecahedron.

The subgroups you've identified are two copies of the dihedral subgroups $D_3$ and $D_2$ respectively. There are many copies of the cyclic and dihedral groups, each corresponding to rotations about a different axis.

For more on this the keyword is the McKay correspondence. These finite groups also show up in Galois theory because each of these finite groups $G$ acts on $\mathbb{C}(t)$ by automorphisms, and so we get $\mathbb{C}(t)$ as a Galois extension of $\mathbb{C}(t)^G$ with Galois group $G$.