Abstract
Isotopies and autotopies of n-ary groups are described. As a consequence, we obtain various characterizations of the group of automorphisms of n-ary groups. We also determine the number of automorphisms of a given n-ary group.
Highlights
Polyadic groups are a natural generalization of classical groups
In this article we find a criterion for when two n-ary groups are isotopic
We characterize the autotopies of such groups and determine the number of all automorphisms of n-ary groups
Summary
Polyadic groups (called n-ary groups) are a natural generalization of classical groups. The first important paper on n-ary groups was written (under the inspiration of Emmy Nother) by Dornte in 1928 (cf [2]). Another important publication is the Post’s article [16] which shows strong relationships with classical groups and gives many significant differences between n-ary groups (n > 2) and ordinary groups. Criteria were found to determine if the groups are isomorphic (for example, [8,9]). Little is known about the automorphisms and autotopies of n-ary groups (cf [14]). The number of autotopies of a given n-ary group (quasigroup) is very large. We only know the exact number of autotopies
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