On autotopies and automorphisms of n-ary linear quasigroups

Abstract

In this article we study structure of autotopies, automorphisms, autotopy groups and automorphism groups of nary linear quasigroups. We find a connection between automorphism groups of some special kinds of n-ary quasigroups (idempotent quasigroups, loops) and some isotopes of these quasigroups. In binary case we find more detailed connections between automorphism group of a loop and automorphism group of some its isotope. We prove that every finite medial n-ary quasigroup of order greater than 2 has a nonidentity automorphism group. We apply obtained results to give some information on automorphism groups of n-ary quasigroups that correspond to the ISSN code, the EAN code and the UPC code.