Around the Hosszú–Gluskin theorem for [formula omitted]-ary groups

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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 used for evaluation how many different n-ary groups (up to isomorphism) exist on some small sets. Moreover, we sketch as the mentioned theorem can also be used for investigation of Q -independent subsets of semiabelian n-ary groups for some special families Q of mappings.