Large Multiplicative Group of Hamel Functions

We construct a multiplicative group of cardinality continuum that is contained (with the exception of the constant function equal to one) in the class of Hamel functions and prove that there is no such group of greater cardinality. Additionally, we answer in the negative \cite[Question 1]{lpss} whether there exists a free group of the size greater than \(\cont\) with respect to the operation of composition that is contained (with the exception of the identity function) in the class of Hamel functions. We also prove the nonexistence of such large free groups with respect to other operations on functions (addition, multiplication).