On the multiplicative groups of free and free commutative quasigroups

We study the structure of the multiplicative groups for the relatively free algebras of some quasigroup varieties, in particular, for the varieties of all quasigroups, all commutative quasigroups, and all TS-quasigroups. In all these cases, the corresponding multiplicative group is free. The work is dedicated to the glorious memory of Prof. Alexandr Alexandrovich Nechaev, who actively explored the possibility of application of quasigroups in cryptography during recent years.