Around the Hosszú-Gluskin theorem for $n$-ary groups

Details Around the Hosszú-Gluskin theorem for $n$-ary groups Around the Hosszú-Gluskin theorem for $n$-ary groups

2005-10-10

arxiv.org/abs/math/0510185v1

Discrete Mathematics 308 (2008), 4861-4876

Mathematics

Group Theory

Scientific paper

10.1016/j.disc.2007.09.005

We survey results related to the important Hossz\'u-Gluskin Theorem on $n$-ary groups adding also 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 Hossz\'u-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 be also used for investigation of $\mathcal{Q}$-independent subsets of semiabelian $n$-ary groups for some special families $\mathcal{Q}$ of mappings.

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