Permutations under Complex Numbers

Question

The question stands:

Let $S=\mathbb{C}-\{1,0\}$. Describe the subgroup of $\operatorname{Perm}(S)$ generated by the functions: $f:S\rightarrow S, z\mapsto 1-z$ and $g:S\rightarrow S, z\mapsto 1/z$.

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I'm having trouble gathering such permutations under $S$ which consist of the complex numbers. Any assistance would be great !

Thank you in advance.

Answer 1

Hints:

Well, to begin with, both $\,f,g\,$ are involutions:

$$f^2(z):=f(1-z):=1-(1-z)=z\;,\;\;g^2(z):=g\left(\frac{1}{z}\right):=\frac{1}{\frac{1}{z}}=z$$

Next:

$$fg(z):=f\left(\frac{1}{z}\right):=1-\frac{1}{z}\;,\;\;gf(z):=g(1-z):=\frac{1}{1-z}\Longrightarrow$$

$$gfg(z)=g\left(1-\frac{1}{z}\right):=\frac{1}{1-\frac{1}{z}}=\frac{z}{z-1}$$

$$fgf(z)=f\left(\frac{1}{1-z}\right)=1-\frac{1}{1-z}=\frac{z}{z-1}\ldots$$

Answer 2

\begin{align} \operatorname{id}(z) & = z \\[10pt] f(z) & = 1-z \\[10pt] g(z) & = \frac1z \\[10pt] f(g(z)) & = \frac{z-1}{z} \\[10pt] g(f(z)) & = \frac{1}{1-z} \\[10pt] f(g(f(z))) & = \frac{z}{1-z} \\[10pt] \end{align}

Now go through the possible compositions of these six functions and observe that you never get any more functions besides thes six, and also that every function on this list has an inverse function that is also on this list.

Answer 3

The elements of our group is words in the letters $f$ and $g$. For starters, notice that $g(g(z))=z$ and $f(f(z))=z$ so we can ignore pairs of the same letter. Our elements are now alternating strings of $f$ and $g$. The identity is the empty string. Now do they commute? $f(g(z))=1-\frac 1z, g(f(z))=\frac 1{1-z}$, no but $g(f(g(z)))=\frac z{z-1}$ which gives us $f(g(f(g(z))))=1-\frac z{z-1}=\frac 1{1-z}=g(f(z))$. This gives us that $f(g(f(z)))=g(f(g(z)))$ and any word can be reduced to no more than three applications of a function. So our group consists of $\{\emptyset,f,g,fg,gf,fgf\}$ with concatenation the operation and $f^2=g^2=\emptyset, fgf=gfg$

Answer 4

@eXtremity From the previous answers you must be able to deduce that your subgroup is isomorphic to S$_3$, the symmetric group on three letters.