Characterization of groups of involutions by means of composite functional equations in two variables (Q2064767)

In this short note the following result is proved. Given an arbitrary group \((G,\cdot)\) the following conditions are pairwise equivalent: (1) \(G\) is involutive; (2) The functional equation \( f(xf(y)) = f(f(x))y^{-1} \text{ for all } x,y\in G \) admits a solution \(f:G\to G\); (3) The functional equation \( f(xf(y)) = y^{-1}f(f(x)) \text{ for all } x,y\in G \) admits a solution \(f:G\to G\). Moreover, when any one of these conditions holds true, the group is necessarily abelian and \(f(x)=cx\) for a certain \(c\in G\). \par Although the proof is quite simple, the result is -- in my opinion -- interesting and very well motivated.