What Is Research?

July 7, 2009

Extensible automorphisms problem solved long ago

Filed under: My research — vipulnaik @ 10:00 pm

Some friends as well as other regular readers of this blog may remember that I mentioned the extensible automorphisms problem: if an automorphism of a group can be extended to any group containing it, must that automorphism be inner?

I had managed to prove this for finite groups, and had drafted a short paper with that and some other related results. My advisor George Glauberman then shared the results of this paper with some other mathematicians, and one of these people, Avinoam Mann, recalled that these problems had been the subject of papers in 1987 and 1990. You can see the extensible implies inner page on the groupprops wiki for more details.

Since I came up with the question myself, and also came up with the final answer myself (of course, with help and guidance from many others), I actually feel vindicated rather than distressed that this question was considered and solved long ago — it shows that the questions I ask have been asked by other people before, and been considered worth publishing.

In mathematics, one may be tempted to feel that it is getting harder and harder to get original results because more and more of the simple stuff has been taken. At the same time, as we acquire more and more knowledge, new researchers can stand on the shoulders of their predecessors and therefore ask and answer deeper questions. I feel that the benefits of being able to look farther and deeper outweigh the costs of having all the more basic results “already taken”.

I still do have a few additional results that I believe to be original, such as this one and its corollary, this one. I also have some partial results on the NPC conjecture, that I believe are new, such as this and others discussed on the NPC conjecture page. I hope to put these together with other results that I am yet to discover and hobble together something publishable in the not-too-distant future.


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