A note on factorizations of finite groups

Document Type : Research Article


Department of Mathematics, University of California Berkeley, CA 94720-3840 USA.


In Question 19.35 of the Kourovka Notebook, M. H. Hooshmand asks whether, given a finite group $G$ and a factorization $\mathrm{card}(G)= n_1\ldots n_k,$ one can always find subsets $A_1,\ldots,A_k$ of $G$ with $\mathrm{card}(A_i)=n_i$ such that $G=A_1\ldots A_k;$ equivalently, such that the group multiplication map $A_1\times\ldots\times A_k\to G$ is a bijection. We show that for $G$ the alternating goup on $4$ elements, $k=3,$ and $(n_1,n_2,n_3) = (2,3,2),$ the answer is negative. We then generalize some of the tools used in our proof, and note a related open question.


The Kourovka Notebook, Unsolved problems in group theory, Nineteenth edition. Edited by Victor Mazurov and Evgeny Khukhro, Russian Academy of Sciences Siberian Branch, Sobolev Institute of Mathematics, Novosibirsk,
2018. 248 pp. MR3981599. Readable at ttps://kourovkanotebookorg.files.wordpress.com/2019/12/19tkt-2.pdf and http://math.nsc.ru/~alglog/19tkt.pdf . Website at https://kourovka-notebook.org/ .
R. Bildanov, V. Goryachenko and A. V. Vasilev, Factoring nonabelian finite groups into two subsets, 7 pages, https://arxiv.org/pdf/2005.12003.pdf.
M. H. Hooshmand, Factor subset of finite group, 2014, ttps://mathoverflow.net/questions/155986/factor-subset-of-finite-group.
M. H. Hooshmand, Basic results for an unsolved problem about factorization of finite groups, to appear.
M. H. Hooshmand, f -representatives groups, to appear.
S. Szabo and A. D. Sands, Factoring groups into subsets, Lecture Notes in Pure and Applied Mathematics, 257, CRC Press, 2009, xvi+269 pp. ISBN: 978-1-4200-9046-8. MR2484422. (This work studies factorizations of finite abelian groups.)