On the solvability of regular subgroups in the holomorph of a finite solvable group

Abstract

We exhibit infinitely many natural numbers $n$ for which there exists at least one insolvable group of order $n$, and yet the holomorph of any solvable group of order $n$ has no insolvable regular subgroup. We also solve Problem 19.90 (d) in the Kourovka notebook.

revised based on referees' and editor's comments