### $(-1)$-enumeration of self-complementary plane partitions.

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It is proved that if a finite abelian group is factored into a direct product of lacunary cyclic subsets, then at least one of the factors must be periodic. This result generalizes Hajós's factorization theorem.

This paper is a contribution to the general tiling problem for the hyperbolic plane. It is an intermediary result between the result obtained by R. Robinson [Invent. Math.44 (1978) 259–264] and the conjecture that the problem is undecidable.