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  • Author or Editor: S. Szabó x
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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.

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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 graphsΓ n has a 2 n size clique. The 4 n vertices ofΓ n 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 ofΓ n forn≤5.

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The paper analyses the views of Ottokár Prohászka, the greatest Hungarian Catholic philosopher of the 20th century on the relationship between labour and social progress, as well as between labour and individual development. The work of Prohászka is seen as a synthesis of modern scientific findings and the Christian world view, which, at the same time, offered a thorough analysis and criticism of the social philosophy of Karl Marx. The author discusses Prohászka's ideas concerning the links between the objective tendencies of social development and the evangelical spirit, and his attempts to elaborate an early version of the theology of labour. Comparing citations, Prohászka's work is valued also for anticipating the ideas in the social encyclics of the popes written several decades later.

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