On Post-Gluskin-Hosszu Theorem A. Gal’mak, V. Balan, G. Vorobiev Abstract. The Post-Gluskin-Hosszu Theorem (also called Gluskin-Hosszu or Hosszu-Gluskin Theorem) refers to an n-ary group 〈A, [ ]〉 and a bi- nary group 〈A, ◦〉, defined on the same set A. E. Post stated and proved this Theorem, while considering instead of the group 〈A, ◦〉, the isomor- phic to it associated group A 0 . This reveals Post’s basic contribution, and justifies the inclusion of his name as leading co-author of the Theorem. Apparently, M. Hosszu was not aware of Post’s result, while L.M. Gluskin did not directly address n-ary groups in his research, focusing mainly on a large class on algebraic systems (positional operatives), for which he obtained a series of notable results, out of these, one of the consequences being exactly the Post-Gluskin-Hosszu Theorem. M.S.C. 2010: 08A02, 20N15, 17A42, 20M10. Key words: groups; n-ary groups; automorphisms. 1 Introduction According to W. D¨ornte [2], we call n-ary group (n ≥ 2) an universal algebra 〈A, []〉 with a given n-ary operation, [ ]: A n → A, which is associative, i.e., for all i ∈ 1,n − 1 in A there holds the associativity condition [[a 1 ...a n ]a n+1 ...a 2n-1 ]=[a 1 ...a i [a i+1 ...a i+n ]a i+n+1 ...a 2n-1 ], and for all i = 1,n and all a 1 ,...,a i-1 ,a i+1 ,...,a n , b ∈ A, the following equations is solvable in A: [a 1 ...a i-1 x i a i+1 ...a n ]= b. We note that D¨ornte’s definition leads, for n = 2, to the usual definition of a binary group. One can identify n-ary groups within the class of all universal algebras in various ways. A rather natural procedure is to point out first, among all the universal algebras, an algebra with an associative n-ary operation, and further to employ the following result: Theorem 1.1. Given a universal algebra 〈A, []〉 endowed with an n-ary associative operation, the following statements are equivalent: Applied Sciences, Vol. 16, 2014, pp. 11-22. c  Balkan Society of Geometers, Geometry Balkan Press 2014.