Kourkovka Notebook: Open problems in group theory

Every once in a while I spot a true gem on the arXiv. Unsolved Problems in Group Theory: The Kourkovka Notebook is such a gem: it is a huge collection of open problems in group theory. Started in 1965, this 19th volume contains hundreds of problems posed by mathematicians around the world. Additionally, problems solved from past volumes are also included with references.

For example, F.M. Markel proved that if $G$ is a finite supersolvable group with no two conjugacy classes having the same number of elements, then $G$ is actually isomorphic to the symmetric group $S_3$. Pretty cool right? Jiping Zhang extended this theorem by replacing ‘supersolvable’ by just ‘solvable’. Problem 16.3 in the Kourkovka notebook asks the obvious: if $G$ is any finite group where no two conjugacy classes have the same size, is $G\cong S_3$? There are of course many more problems of varying technicality, but there should be something in here for any group theorist.

I’ve always thought that you can gauge the health of a discipline by the quality of open problems in it. If that’s true, then the Kourkovka notebook shows that group theory is thriving very well.

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