Generating free groups by loop translations (Q1902963)

It is well known that to every quasigroup \(\mathcal Q\) is subjoined a multiplication group which is generated by all left and right translations of \(\mathcal Q\). The importance of multiplication groups lies in the connection between the structures of a quasigroup and its multiplication group. Conversely, to determine which permutation group can be a multiplication group of some quasigroup is an unresolved problem. In this paper, the author constructs a commutative loop (i.e. quasigroup with identity element) such that its multiplication group is isomorphic to a given free group of finite rank.