TITLE: What is the order of $f(x) = \frac{2}{2-x}$ as a permutation?
QUESTION [1 upvotes]: $f(x) = \frac{2}{2-x}$ defines a permutation of the set $A={\bf R}\setminus \{0,1,2\}$. What is its order in the permutation group $S_A$?
I don't know how to find orders of functions. Please help!

REPLY [3 votes]: For the purpose at hand, tomasz's answer resolves the OP's question, but I thought I'd just add a note that there is a "smart" way to compute the order of a function if it's a linear fractional transformation of the form $f(x)=(ax+b)/(cx+d)$ with real coefficients:  compute the eigenvalues of the $2\times2$ matrix
$$\pmatrix{a&b\\c&d}$$
If they are of the form $\lambda=re^{\pm\pi im/n}$ with $r$ real and the fraction $m/n$ in reduced form (i.e., $\gcd(m,n)=1$), then the function has order $n$.  (That is, $f(f(\cdots f(x)))=x$, where there are $n$ $f$'s on the left hand side.)
In the OP's case, the matrix is
$$\pmatrix{0&2\\-1&2}$$
and the eigenvalues are $\sqrt2e^{\pm\pi i/4}$, so the order is $4$.
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