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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.
Old-fields (44, aged 1–15 years, from Czech Republic and Hungary) were sorted according to their soil moisture and nitrogen content into wet, mesic or dry, and nutrient poor, moderate or nutrient rich categories, resulting in 8 combinations (dry and nutrient rich fields were not present). The vegetation of old fields was sampled using phytosociological relevès. The changes in species cover data and importance of species trait categories were analysed in relation to three environmental factors, i.e., time since abandonment, soil moisture and total soil nitrogen using ordination, generalized linear models (GLM) and regression tree methods. Successional seres in the first 15 years after field abandonment were divergent. Species diversity significantly decreased with increasing site moisture and was highest in sites with moderate nitrogen content; while the relationship with time was not significant. Raunkiaer life forms and life strategies (sensu Grime) were generally the most predictive species traits considering species occurrence during the course of succession, the type of dispersal considering the different moisture status, and the ability to lateral spread considering the nutrient status of the old-fields. Most trends appeared in both parametric GLM and non-parametric regression tree analyses, several only in GLM. We consider regression trees to be a more convenient tool than GLM in cases such as ours with a rather small number of samples and robust character of data. Another advantage is that a hierarchy of species traits is taken into account. Thus, the occurrence of a species along an environmental gradient can be predicted if the species possesses a certain combination of traits.
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.
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.