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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2018. Vol. 26

Automorphisms of some magmas of order k + k2

Author(s)
A. V. Litavrin
Abstract

This paper is devoted to the study of automorphisms of finite magmas and to the representation of the symmetric permutation group Sk and some of its subgroups by automorphism groups of finite magmas. The theory that studies automorphism groups of magmas is well developed and is represented by a multitude of works, when magma is a quasigroup, semigroup, loop, monoid or group. There are also studies in which problems related to the study of automorphisms of magmas that are not a semigroup or quasigroup are considered.

In this paper, we introduce some finite magmas 𝕾 = (V, ∗) of order k + k2. For magma 𝕾  it was possible to describe the automorphism group and write down the general form of the automorphism. In addition, the connection between automorphisms of magmas 𝕾  and permutations of a finite set of k elements has been revealed. All automorphisms of magma 𝕾  are parametrized by permutations from a certain subgroup (a description of this subgroup is given) of the symmetric permutation group Sk. 

In addition, it is established that the group Sk is isomorphic to the group of all automorphisms Aut (𝕾 ) of a suitable magma 𝕾 of order k + k2.

About the Authors

Andrey V. Litavrin, Cand. Sci. (Phys.–Math.), Siberian Federal University, 79, Svobodny pr., Krasnoyarsk, 660041, Russian Federation, e-mail: anm11@rambler.ru

For citation
Litavrin A.V. Automorphisms of some magmas of order k + k2. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 26, pp. 47-61. (in Russian) https://doi.org/10.26516/1997-7670.2018.26.47
Keywords
automorphisms of a magma, automorphisms of a groupoid, groups of automorphisms
UDC
512.54+512.57
MSC
17B40, 17B30
DOI
https://doi.org/10.26516/1997-7670.2018.26.47
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