In the years leading up to my Ph.D., I considered a few problems in group theory where I either got partial results or didn’t manage to make headway. In some cases, I solved the problems completely but they weren’t important enough to be worth publishing.
Please also consider checking out the Group Properties Wiki, a handy reference for group theory that I worked on during my Ph.D. study years.
- Extensible automorphisms problem: I came up with the question and solved this problem based on some existing research, but later discovered that I’d been scooped by about 20 years.
- Potentially characteristic subgroups characterization problem: I came up with the question and solved the case for general groups some time in 2009. My solution can be viewed here. To my knowledge, both the question and my answer are original, but the proof is not too deep and I didn’t consider it worth writing up for publication. Analogous results fail for p-groups, and the proof of this uses somewhat nontrivial results.
- Some unpublished research, conducted along with John Wiltshire-Gordon, then an undergraduate at the University of Chicago, and now a graduate student at the University of Michigan. Some parts of this were related to the section on “Baer correspondence up to isoclinism” in my thesis. When I get the time, I will put up links from this page to a draft of the research we did, along with links to more detailed explanations.
- Slight extensions of results of Jonah and Konvisser, that may or may not be original (a literature review and discussions with my advisor suggested my extensions may be original, but I didn’t get time to investigate this thoroughly, and the extensions were too minor to be worth writing up). I will link to wiki pages where the extensions are stated.