Geodesic
Factoring nonabelian finite groups into two subsets
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 683-689

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A group $G$ is said to be factorized into subsets $A_1, A_2,$ $\ldots, A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. We consider the following conjecture: for every finite group $G$ and every factorization $n=ab$ of its order, there is a factorization $G=AB$ with $|A|=a$ and $|B|=b$. We show that a minimal counterexample to this conjecture must be a nonabelian simple group and prove the conjecture for every finite group the nonabelian composition factors of which have orders less than $10 000$.
Keywords: factoring of groups into subsets, finite group, finite simple group, maximal subgroups. 
@article{SEMR_2020_17_a10,
     author = {R. R. Bildanov and V. A. Goryachenko and A. V. Vasil'ev},
     title = {Factoring nonabelian finite groups into two subsets},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {683--689},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a10/}
}
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R. R. Bildanov; V. A. Goryachenko; A. V. Vasil'ev. Factoring nonabelian finite groups into two subsets. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 683-689. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a10/