Geodesic
On Post--Gluskin--Hosszu theorem
Problemy fiziki, matematiki i tehniki, no. 1 (2013), pp. 55-60.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Mots-clés : group, automorphism.
Keywords: $n$-ary group 
@article{PFMT_2013_1_a9,
     author = {A. M. Gal'mak and G. N. Vorobiev},
     title = {On {Post--Gluskin--Hosszu} theorem},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {55--60},
     publisher = {mathdoc},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2013_1_a9/}
}
TY  - JOUR
AU  - A. M. Gal'mak
AU  - G. N. Vorobiev
TI  - On Post--Gluskin--Hosszu theorem
JO  - Problemy fiziki, matematiki i tehniki
PY  - 2013
SP  - 55
EP  - 60
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PFMT_2013_1_a9/
LA  - ru
ID  - PFMT_2013_1_a9
ER  - 
%0 Journal Article
%A A. M. Gal'mak
%A G. N. Vorobiev
%T On Post--Gluskin--Hosszu theorem
%J Problemy fiziki, matematiki i tehniki
%D 2013
%P 55-60
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PFMT_2013_1_a9/
%G ru
%F PFMT_2013_1_a9
A. M. Gal'mak; G. N. Vorobiev. On Post--Gluskin--Hosszu theorem. Problemy fiziki, matematiki i tehniki, no. 1 (2013), pp. 55-60. http://geodesic.mathdoc.fr/item/PFMT_2013_1_a9/

[1] W. Dörnte, “Untersuchungen über einen verallgemeinerten Gruppenbegrieff”, Math. Z., 29 (1928), 1–19 | MR

[2] E. L. Post, “Polyadic groups”, Trans. Amer. Math. Soc., 48:2 (1940), 208–350 | MR

[3] V. I. Tyutin, “K aksiomatike $n$-arnykh grupp”, Dokl. AN BSSR, 29:8 (1985), 691–693 | MR | Zbl

[4] A. M. Galmak, “Ob opredelenii $n$-arnoi gruppy”, Mezhdunar. konf. po algebre, Tez. dokl., Novosibirsk, 1991, 30

[5] A. M. Galmak, $n$-Arnye gruppy, v. 1, GGU im. F. Skoriny, Gomel, 2003, 196 pp.

[6] S. A. Rusakov, Algebraicheskie $n$-arnye sistemy, Navuka i tekhnika, Mn., 1992, 245 pp.

[7] A. M. Galmak, $n$-Arnye gruppy, v. 2, Izd. Tsentr BGU, Minsk, 2007, 323 pp.

[8] L. M. Gluskin, “Pozitsionnye operativy”, Mat. sbornik, 68 (110):3 (1965), 444–472 | MR | Zbl

[9] M. Hosszu, “On the explicit form of $n$-group operations”, Publ. Math., 10:1–4 (1963), 88–92 | MR

[10] E. I. Sokolov, “O teoreme Gluskina–Khossu dlya $n$-grupp Dernte”, Mat. issledovaniya, no. 39, 187–189 | MR