On some questions concerning a functional equation involving Möbius transformations (Q1587845)

It is proved that any bijection of a field \({\mathbb F}\) onto itself which preserves the operation \((x,y) \to (x+y)/(x-y)\) is necessarily the identity mapping whenever \({\mathbb F} = {\mathbb Q}, ~{\mathbb R}, ~{\mathbb F}_p ~(p \not= 5)\) or \({\mathbb F}\) is a Galois extension of \({\mathbb Q}\) of degree \(2^k\). This statement is no longer true for \({\mathbb F}_5\).