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.


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


Zbl 0496.05053
Full Text: DOI


[1] Bruck, R.H, Finite nets, I, numerical invariants, Canad. J. math., 3, 94-107, (1951) · Zbl 0042.38802
[2] Chang, G.J; Hsu, D.F; Rogers, D.G, Additive variations on a graceful theme: some results on harmonious and other related graphs, (), 181-197 · Zbl 0496.05053
[3] Friedlander, R.J; Gordon, B; Miller, M.D, On a group sequencing problem of ringel, (), 307-321
[4] Gorenstein, D, Finite groups, (1968), Harper and Row New York · Zbl 0185.05701
[5] Graham, R.L; Sloane, N.J.A, On additive bases and harmonious graphs, SIAM J. algebraic discrete methods, 4, 382-404, (1980) · Zbl 0499.05049
[6] Gumm, H.P, Encoding of numbers to detect typing errors, Int. J. appl. engng. ed., 2, 61-65, (1986)
[7] Hall, M; Paige, L.J, Complete mappings of finite groups, Pacific J. math., 5, 541-549, (1955) · Zbl 0066.27703
[8] Mann, H.B, The construction of orthogonal Latin fields, Ann. math. statist., 13, 418-423, (1942) · Zbl 0060.02706
[9] Paige, L.J, Neofields, Duke math. J., 16, 39-60, (1949) · Zbl 0040.30501
[10] Ringel, G, Cyclic arrangements of the elements of a group, Notices amer. math. soc., 21, A95-A96, (1974)
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