Kourovka Notebook Problem 21.97

Kourovka Notebook Problem $21.97$ asks whether every positive rational number can be realized as $|\operatorname{Aut}(G)|/|G|$ for some finite group $G$. The cited recent work explicitly says that the question for all finite groups remains open.

March 31, 2026: finite abelian groups

A preprint claims that for finite abelian groups, if $|\operatorname{Aut}(G)|/|G|=a/b$ in lowest terms, then $b$ is squarefree and no odd prime occurs as the ratio; it also proves that every power of $2$ does occur. Thus it gives substantial restrictions and realizations in the abelian case, but does not resolve Problem $21.97$ for arbitrary finite groups.

Current status (as of August 2026): The finite-abelian case has a claimed preprint result, while the existence question for all finite groups remains open.