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 \(n\)-ary 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 non-identity 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.