Denote by the number of elements in satisfying . Frobenius’ theorem states that if is a divisor of , then is a multiple of .
In the article On an inverse Problem to Frobenius’ theorem (2011, Springerlink) Wei Meng and Jiangtao Shi propose the following problem.
Let be a positive integer. Classify all groups with the property that for every divisor of .
Denote by the largest positive integer such that for all divisors of . It is a standard result that if and only if is cyclic. Wei Meng and Jiangtao Shi have classified groups with and those with (with Kelin Chen).
Here are some of my thoughts about the general problem. Let be a group and suppose that for all . Let be a prime divisor of and let be a Sylow -subgroup of (of order ). Then
1) If , then is normal and cyclic.
2) If , then is normal or is cyclic.
3) If for , then is normal or
4) If , then is normal or else is cyclic and .
Denote by the number of Sylow -subgroups of . The facts above follow from a theorem of G. A. Miller which states that if , then and if , then .
Applying this we see that when for all , every Sylow subgroup of is cyclic or normal. Furthermore, any Sylow -subgroup is central if . To see this, note that is centralized by -sylows for since they are normal. The subgroup is also centralized by -sylows for since and the automorphism group of cyclic group of order has order coprime to . Hence by Burnside's normal complement theorem , where is cyclic and coprime to , and , reducing the classification of such to groups with order of the form .
A good starting point for classifying groups with for all might be to show that , where is cyclic and coprime to primes that are small enough. However, it seems difficult to find a good upper bound for the number of Sylow -subgroups in this case. Applying 1), 2), 3), 4) or Miller's theorem no longer works here, so improving Miller's result is one possibility. It might be true that . This is just a guess I made by looking at groups of order and values of when .
We would also need an upper bound for the size of primes dividing the order of the automorphism group of a -sylow . By 3), this would be a 2-group of order with a cyclic subgroup of order . I believe these are classified, so it would be a matter of computing the automorphism groups and thus giving an upper bound for the size of primes dividing . Then -sylows would centralize any -sylow for greater than any prime divisor of .