Automorphisms of \(n\)-ary groups. (English) Zbl 1497.20066

Dudek, Wieslaw A.

Result. Math. 77, No. 1, Paper No. 46, 16 p. (2022).

By the Hosszú-Gluskin theorem every \(n\)-group (\(n > 2\)) \((G,f)\) can be derived from a binary group \((G, \cdot )\), an automorphism \(\varphi \) of \((G, \cdot )\) and an element \(b \in G\) such that \(\varphi (b) = b\), denote \((G,f) = \mathrm{der}_{\varphi ,b}(G, \cdot )\). Here, automorphisms and autotopies of \(n\)-ary groups are studied using this fact. Bijections \({\alpha _1},\dots ,{\alpha _n},\delta \) from an \(n\)-groupoid \((G,f)\) to an \(n\)-groupoid \((H,g)\) are an isotopy between \((G,f)\) and \((H,g)\) iff \(\delta (f({x_1},\dots ,{x_n})) = g({\alpha _1}{x_1},\dots ,{\alpha _n}{x_n})\). For instance, it is shown that groups of autotopies of isotopic \(n\) -groups are isomorphic, number of autotopies of \(n\)-group \((G,f)\) is \(|G{|^n} \cdot |\operatorname{Aut}(G, \cdot )|\) and the number of automorphisms of a derived \(n\)-group \(\mathrm{der}_{\varphi ,b}(G, \cdot )\) is expressed using properties of the binary group \((G, \cdot )\).

Reviewer: Jaak Henno (Tallinn)

MSC:

20N15

\(n\)-ary systems \((n\ge 3)\)

Keywords:

autotopy; isotopy; automorphism; \(n\)-ary group

References:

[1]	Belousov, V.D.: \(n\)-Ary Quasigroups. Sţiinţa, Chişinǎu (1972). (Russian)
[2]	Dörnte, W., Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z., 29, 1-19 (1928) · JFM 54.0152.01 · doi:10.1007/BF01180515
[3]	Dudek, WA, Remarks on \(n\)-groups, Demonstr. Math., 13, 165-181 (1980) · Zbl 0447.20052
[4]	Dudek, WA, Idempotents in \(n\)-ary semigroups, Southeast Asian Bull. Math., 25, 97-104 (2001) · Zbl 0986.20068 · doi:10.1007/s10012-001-0097-y
[5]	Dudek, WA, Varieties of polyadic groups, Filomat, 9, 657-674 (1995) · Zbl 0860.20056
[6]	Dudek, WA; Głazek, K., Around the Hosszú-Gluskin Theorem for \(n\)-ary groups, Discrete Math., 308, 4861-4876 (2008) · Zbl 1153.20050 · doi:10.1016/j.disc.2007.09.005
[7]	Dudek, W.A., Głazek, K., Gleichgewicht, B.: A note on the axioms of \(n\)-groups. Colloquia Math. Soc. J. Bolyai bf 29 “Universal Algebra”, Esztergom (Hungary), (1977), 195-202. North-Holland, Amsterdam (1982) · Zbl 0525.20059
[8]	Dudek, WA; Michalski, J., On a generalization of Hosszú theorem, Demonstr. Math., 15, 783-805 (1982) · Zbl 0523.20045
[9]	Dudek, WA; Michalski, J., On retracts of polyadic groups, Demonstr. Math., 17, 281-301 (1984) · Zbl 0573.20067
[10]	Gorkunov, EV; Krotov, DS; Potapov, VN, On the number of autotopies of an \(n\)-ary quasigroup of order 4, Quasigroups Relat. Syst., 27, 227-250 (2019) · Zbl 1435.20081
[11]	Gluskin, L.M.: Positional operatives. Mat. Sbornik (N.S.) 68(110), 444-472 (1965). (Russian) · Zbl 0244.20092
[12]	Hillar, ChJ; Rhea, DL, Automorphisms of finite abelian groups, Am. Math. Mon., 114, 917-923 (2007) · Zbl 1156.20046 · doi:10.1080/00029890.2007.11920485
[13]	Hosszú, M., On the explicit form of \(n\)-group operations, Publ. Math. Debrecen, 10, 88-92 (1963) · Zbl 0118.26402
[14]	Khodabandeh, H.; Shahryari, M., On the automorphisms and representations of polyadic groups, Commun. Algebra, 40, 2199-2212 (2012) · Zbl 1257.20074 · doi:10.1080/00927872.2011.576738
[15]	Marini, A.; Shcherbacov, V., On autotopies and automorphisms of \(n\)-ary linear quasigroups, Algebra Discrete Math., 2, 59-83 (2004) · Zbl 1067.20085
[16]	Post, EL, Polyadic groups, Trans. Am. Math. Soc., 48, 208-350 (1940) · JFM 66.0099.01 · doi:10.1090/S0002-9947-1940-0002894-7
[17]	Shcherbacov, V., Elements of Quasigroups and Applications (2017), CRC Press: Taylor & Francis Group, CRC Press · Zbl 1491.20003 · doi:10.1201/9781315120058