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Harmonious groups. (English) Zbl 0718.20013
A finite group G is called harmonious if the elements of G can be listed \(g_ 1,g_ 2,...,g_ n\) so that \(G=\{g_ 1g_ 2,g_ 2g_ 3,...,g_{n-1}g_ n,g_ ng_ 1\}\). The main result of this paper is the following theorem: If G is a finite, non-trivial Abelian group, then G is harmonious if and only if G has a non-cyclic or trivial Sylow 2- subgroup and G is not an elementary 2-group (Theorem 6.6). In section 4 of the paper, it is shown that if finite groups G and H are harmonious and H has odd order, then \(G\times H\) is harmonious (Theorem 4.1). This result completes the characterization of elegant cycles begun by G. J. Chang, D. F. Hsu and D. G. Rogers [Congr. Numerantium 32, 181-197 (1981; Zbl 0496.05053)]. In the final section of the paper, the authors also define and investigate harmonious-matched groups.

MSC:
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F65 Geometric group theory
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