@article { author = {Bergman, G. M.}, title = {A note on factorizations of finite groups}, journal = {Journal of the Iranian Mathematical Society}, volume = {1}, number = {2}, pages = {157-161}, year = {2020}, publisher = {Iranian Mathematical Society}, issn = {2717-1612}, eissn = {2717-1612}, doi = {10.30504/jims.2020.108338}, abstract = {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.}, keywords = {Factorization of a finite group,Product of subsets}, url = {https://jims.ims.ir/article_108338.html}, eprint = {https://jims.ims.ir/article_108338_8df75833b842c30981a08daf42636a75.pdf} }