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We survey the results related to the important Hosszú–Gluskin Theorem on n-ary groups adding several new results and comments. The aim of this paper is to write all such results in uniform and compressive forms. Therefore some proofs of new results are only sketched or omitted if their completing seems to be not too difficult for readers. In particular, we show as the Hosszú–Gluskin Theorem can be...
If H is a subgroup of the symmetric group S 4 , then a 3-groupoid (S,f) is called H-permutable if (x σ(1) ,x σ(3) ) = x σ(4) (x 1 ,x 2 ,x 3 ) = x 4 for every σ H. Some classes of generalized idempotent H-permutable 3-groupoids are equivalent to Steiner, Mendelsohn and other quadruple...
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