Basic results on an unsolved problem about factorization of finite groups

Abstract

We deal with an unsolved problem in group theory (question 19.35 of Kourovka Notebook). It asks whether, given a finite group G and a factorization one can always find subsets such that and The case k = 2 is of special interest. We prove that the answer is positive for all finite supersolvable (and so abelian) groups. However, G. Bergman has recently given a counterexample to the question for the case k = 3. Regarding the special case k = 2, at first, we prove that the statement is true for all finite solvable groups, and groups such that for every divisor d of their order, there is a subgroup of order or index d. Thereafter, by proving an extension theorem we show that the answer is positive if it is true for all simple groups with no subgroup of prime index. Also, we give some other theorems and criteria, for checking the property and state a question about some candidates for probable counterexamples. We end by noting some related connections, questions and problems about the topic.