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PFMT, 2013, Issue 1(14), Pages 55–60 (Mi pfmt222)  

This article is cited in 8 scientific papers (total in 8 papers)

MATHEMATICS

On Post–Gluskin–Hosszu theorem

A. M. Gal'mak, G. N. Vorobiev

Mogilev State University of Food Technologies, Mogilev

Abstract: Post–Gluskin–Hosszu theorem is known as Gluskin-Hosszu or Hosszu–Gluskin theorem. In the formulation of this theorem there is an $n$-ary group $< A, [ ] >$ and some binary group $< A,\circ >$. These groups have a common carrier $A$. E. Post formulated and proved this theorem considering isomorphous copy of $A_0$ (associated group) instead of group $< A,\circ >$. In our opinion the absence of the name of E. Post in the title of the theorem is an embarrassing mistake which must be corrected. Apparently, M. Hosszu didn’t know anything about the results of E. Post. It is necessary to note that L. M. Gluskin was not engaged in the study of $n$-ary groups. He investigated a large class of algebraic systems — positional operatives, and achieved a series of important results. Post–Gluskin–Hosszu theorem is among numerous consequences of these results.

Keywords: group, $n$-ary group, automorphism.

Full text: PDF file (355 kB)
References: PDF file   HTML file
UDC: 512.548
Received: 30.05.2012

Citation: A. M. Gal'mak, G. N. Vorobiev, “On Post–Gluskin–Hosszu theorem”, PFMT, 2013, no. 1(14), 55–60

Citation in format AMSBIB
\Bibitem{GalVor13}
\by A.~M.~Gal'mak, G.~N.~Vorobiev
\paper On Post--Gluskin--Hosszu theorem
\jour PFMT
\yr 2013
\issue 1(14)
\pages 55--60
\mathnet{http://mi.mathnet.ru/pfmt222}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Galmak, N. A. Schuchkin, “K teoreme Posta o smezhnykh klassakh”, Chebyshevskii sb., 15:2 (2014), 6–20  mathnet
    2. A. M. Galmak, N. A. Schuchkin, “Tsiklicheskie $n$-arnye gruppy i ikh obobscheniya”, PFMT, 2014, no. 2(19), 46–53  mathnet
    3. F. M. Malyshev, “The Post-Gluskin-Hosszú theorem for finite $n$-quasigroups and self-invariant families of permutations”, Sb. Math., 207:2 (2016), 226–237  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. V. Cheremushkin, “Analogues of Gluskin–Hosszú and Malyshev theorems for strongly dependent $n$-ary operations”, Discrete Math. Appl., 29:5 (2019), 295–302  mathnet  crossref  crossref  isi  elib
    5. A. V. Cheremushkin, “Obobschenie teorem Gluskina–Khossu i Malysheva na sluchai cilno zavisimykh $n$-arnykh operatsii”, PDM. Prilozhenie, 2018, no. 11, 23–25  mathnet  crossref
    6. F. M. Malyshev, “Slabo obratimye $n$-kvazigruppy”, Chebyshevskii sb., 19:2 (2018), 304–318  mathnet  crossref  elib
    7. A. V. Cheremushkin, “Teorema Posta dlya silno zavisimykh $n$-arnykh polugrupp”, Diskret. matem., 31:2 (2019), 152–157  mathnet  crossref  elib
    8. A. V. Cheremushkin, “Svoistva cilno zavisimykh $n$-arnykh polugrupp”, PDM. Prilozhenie, 2019, no. 12, 36–41  mathnet  crossref
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