On multiplication groups of relatively free quasigroups isotopic to abelian groups

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an abelian group, then its multiplication group MltQ is a Frobenius group. Conversely, if MltQ is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an abelian group. If Q is, in addition, finite, then it must be a central quasigroup. A quasigroup Q = Q(·) is usually defined as a binary system in which the equations a · x = b, y · a = b have unique solutions for all a, b ∈ Q. One then sets x = a\b and y = b/a. When one wishes to describe quasigroups by identities, one regards Q = Q(·, /, \) as a set with three binary operations that are interconnected by relations x · (x\y) = y = x\(x · y) and (x · y)/y = x = (x/y) · y. This is the approach we shall take in this paper. Its main result concerns quasigroups that are free in the class of all quasigroups isotopic to abelian groups. We shall show that their multiplication groups are Frobenius groups (i.e., permutation groups that are transitive, but not regular, and where every nonidentity permutation fixes at most one point), and that the stabilizers of these groups (the so called inner mapping groups) are free. This contrasts with another result of ours, which states that a finite quasigroup with Frobenius multiplication group has to be central. ∗Work supported by institutional grant MSM 113200007, and by Grant Agency of Czech Republic, grant number 201/99/0263.