If a finite abelian group is factored into a direct product of its cyclic subsets, then at least one of the factors is periodic.
This is a famous result of G. Hajós. We propose to replace the cyclicity of the factors by an abstract property that still
guarantees that one of the factors is periodic. Then we present applications of this approach.
The statement, that in a tiling by translates of ann-dimensional cube there are two cubes having common (n-1)-dimensional faces, is known as Keller's conjecture. We shall prove that there is a counterexample for this conjecture if and only if the following graphsΓn has a 2n size clique. The 4n vertices ofΓn aren-tuples of integers 0, 1, 2, and 3. A pair of thesen-tuples are adjacent if there is a position at which the difference of the corresponding components is 2 modulo 4 and if there is a further position at which the corresponding components are different. We will give the size of the maximal cliques ofΓn forn≤5.
Rédei's theorem asserts that if a finite abelian group is expressed as a direct product of subsets of prime cardinality, then at least one of the factors must be periodic. (A periodic subset is a direct product of some subset and a nontrivial subgroup.) A. D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality that is a power of a prime, then at least one of the factors is periodic. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subsets of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.
Authors:T. Becze-Deák, L. Bottyán, G. Corradi, L. Korecz, D. Nagy, K. Polgár, S. Sayed and H. Spiering
57Co Mössbauer emission spectra of undoped and Fe or Mg melt-doped LiNbO3 single crystals show substantial amounts of the nucleogenic Fe3+ charge state (*Fe3+) which was generated as an after-effect of the electron-capture of 57Co2+. The proportion of *Fe3+ is markedly dependent on the Mg content and on the stoichiometry of the sample. Electron trapping is described within the model of competing acceptors. The capabilities of the model are investigated in defect structure analysis and charge trapping studies of LiNbO3.