I list below some problems for which solutions are not know to my knowledge. I plan to put up separate files describing where I stand with respect to the problems. I also plan to create a PDF version of this file for easy download.
My own unsolved problems in group theory:
- Extensible automorphisms: Call an automorphism of a group extensible if for any embedding of that group in a bigger group, the automorphism can be lifted to an automorphism of the bigger group. Is every extensible automorphism inner? I plan to put all the progress made so far at the Extensible automorphisms page.
- Normal and potentially characteristic subgroups: Call a subgroup of a group potentially characteristic if there is a group containing the bigger group in which the subgroup is characteristic. Clearly, every potentially characteristic subgroup is normal. Is the converse true? That is, is every normal subgroup potentially characteristic? This question is intimately related to the problem of extensible automorphisms. I plan to put all the progress made so far at the Potentially characteristic subgroups page.
- Cancellation of direct products What conditions are sufficient to ensure that the direct product operation on groups is cancellative? That is, if G X H and G X K are isomorphic, are H and K isomorphic? I have settled the case where G, H and K are all finite, and I hope to settle more cases as I proceed. View a presentation here.
- Strictly characteristic and fully characteristic: A subgroup of a group is said to be fully characteristic or fully invariant if every endomorphism of the group restricts to an endomorphism of the subgroup. A subgroup of a group is said to be strictly characteristic or distinguished if every surjective endomorphism of the group restricts to an endomorphism of the subgroup. Every fully characteristic subgroup of a strictly characteristic subgroup is strictly characteristic. Is being fully characteristic the weakest property that acts as a "transiter" for being strictly characteristic? I plan to put up my writeup on this.
If you are interested in open problems in group theory, check out the Open Problems in Combinatorial Group Theory Page on the CUNY website.
Semistandard problem (adapted by me) in combinatorics and number theory
- The Equal partitions problem is as follows: Let m and n be two integers with m at least 3 and n at least 2. Consider an arbitrary partition of the integers from 1 to mn, into m sets, each of size n. Prove that I can pick one element from each part and divide the elements into two piles, such that the number of elements in both piles differs by at most 1, and the sum of the elements in both piles is equal. I know the solution only for m being 3 or 4.
My own problems in other areas:
- If the product of a space with every paracompact space is paracompact, what can be said about the space? Clearly, compactness is sufficient and paracompactness is necessary. I had earlier wondered whether compactness is necessary but this was disproved in a post on Mathlinks. The proof also provided a concret e property which was sufficient to guarantee that the product with every paracompact space is paracompact. I want to explore further questions like: is this a necessary and sufficient property?
Earlier open problems for which I have obtained solutions: