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

## TL;DR

The paper investigates the conditions under which the holomorph of a solvable group contains insolvable regular subgroups, providing examples and solving a problem from the Kourovka notebook.

## Contribution

It demonstrates that for infinitely many orders, the holomorph of any solvable group lacks insolvable regular subgroups, and solves a specific open problem.

## Key findings

- Existence of infinitely many orders with insolvable groups but no insolvable regular subgroups in their holomorphs.
- Resolution of Problem 19.90 (d) in the Kourovka notebook.
- Insight into the structure of regular subgroups in holomorphs of solvable groups.

## 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.