Automorphisms of finite quasigroups with no subquasigroups

It is shown that polynomially complete quasigroups with no subquasigroups are quasitermal. The case of transitive action of the automorphism group on these quasigroups is considered. In particular the case of quasigroup of a prime power order defined on arithmetic vector space over a finite field is considered in details. There are found some necessary conditions under which a multplication in this space given in terms of coordinates corresponds to a quasigroup. The case of the 2-element field is considered in detalis. In this case the quasigroup multiplication is given in terms of Boolean function. The is found a criteria for a quasigroup multiplication. Under some assumptions there are classified up to an isotopy all quasigroups of order 4 in terms of Boolean function. Polynomially complete quasigroups play a significant role because the problem of solutions of polynomial equation in them is NP-complete. This property is important for the securing information, since crypto-transformations are defined in terms of quasigroup operations. The same argument shows the importance of quasigroups with no proper subquasigroups.