This article is cited in 2 papers

Mathematical logic, algebra and number theory

Factoring nonabelian finite groups into two subsets

R. R. Bildanov^a, V. A. Goryachenko^b, A. V. Vasil'ev<sup>c

a</sup> Specialized Educational Scientific Center of Novosibirsk State University, 11/1, Pirogova str., Novosibirsk, 630090, Russia
^b Novosibirsk State University, 1, Pirogova str., Novosibirsk, 630090, Russia
^c Sobolev Institute of Mathematics, 4, Koptyuga ave., Novosibirsk, 630090, Russia

Abstract: 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.

UDC: 512.542

MSC: 20D60

Received April 30, 2020, published May 21, 2020

Language: English

DOI: 10.33048/semi.2020.17.046

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