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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1166,
author = {S\'andor Szab\'o},
title = {Factoring an odd abelian group by lacunary cyclic subsets},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {30},
year = {2010},
pages = {137-146},
zbl = {1231.20050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1166}
}
Sándor Szabó. Factoring an odd abelian group by lacunary cyclic subsets. Discussiones Mathematicae - General Algebra and Applications, Tome 30 (2010) pp. 137-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1166/
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