Automorphisms of some finite magmas with an order strictly less than the number N(N+1) and a generating set of N elements

A. V. Litavrin

Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 2 (2019), pp. 70-87

Résumé

In this paper, we study the problems of describing automorphism groups of certain finite magmas. Some finite magmas $\mathfrak{G} = (V,*)$, generated by $n$ elements and order $|V| $, satisfying the inequalities $n+1 \le | V |$. Constructed magmas $\mathfrak{G}$ are not semigroups or quasigroups. For the magmas $\mathfrak{G}$, the general form of the automorphism is indicated and the description of the group of all automorphisms is given. It is shown that the group of all automorphisms is isomorphic to a certain subgroup (the description of this group is given) of the symmetric permutation group $S_n$, where $n$ is the number of elements of a suitable generating set of the magma $\mathfrak{G}$. It is proved that every finite cyclic group of order $n\ge 2$ is isomorphic to the group of all automorphisms of the appropriate magma $\mathfrak{G}$. A similar result was obtained for the fourth Klein group. In addition, it was shown that for any finite group $G$ you can choose a suitable magma $\mathfrak{G}$ such that $G$ is isomorphic to some subgroup of $Aut (\mathfrak{G})$ (an algorithm for constructing magma is given $\mathfrak{G}$ for an arbitrary finite group $G$).

Détail

Keywords: magmas, finite magma automorphisms, finite groupoid automorphisms, finite cyclic group, Dihedron group.
Mots-clés : groupoids