Geodesic
Factoring an odd abelian group by lacunary cyclic subsets
Discussiones Mathematicae. General Algebra and Applications, Tome 30 (2010) no. 2, pp. 137-146

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It is a known result that if a finite abelian group of odd order is a direct product of lacunary cyclic subsets, then at least one of the factors must be a subgroup. The paper gives an elementary proof that does not rely on characters.
Keywords: factorization of finite abelian groups, periodic subsets, cyclic subsets, lacunary cyclic subsets, Hajós-Rédei theory 
Szabó, Sándor. Factoring an odd abelian group by lacunary cyclic subsets. Discussiones Mathematicae. General Algebra and Applications, Tome 30 (2010) no. 2, pp. 137-146. http://geodesic.mathdoc.fr/item/DMGAA_2010_30_2_a0/
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[5] S. Szabó and A.D. Sands, Factoring Groups into Subsets, Chapman and Hall, CRC, Taylor and Francis Group, Boca Raton 2009.