Abstract
Let S denote the unit circle on the complex plane and star:S-2 -> S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S,star) is isomorphic to the group (S,center dot); thus, it is a group, where center dot is the usual multiplication of complex numbers. However, an elementary construction of such isomorphism has not been published so far. We present an elementary construction of all such continuous isomorphisms F from (S,center dot) into (S,star) and obtain, in this way, the following description of operation star: x star y=F(F-1(x)center dot F-1(y)) for x,y is an element of S. We also provide some applications of that result and underline some symmetry issues, which arise between the consequences of it and of the analogous outcome for the real interval and which concern functional equations. In particular, we show how to use the result in the descriptions of the continuous flows and minimal homeomorphisms on S.