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.

arXiv commentrevised based on referees' and editor's comments

References in corpus (4)

• Skew braces and the Yang-Baxter equation
• On skew braces (with an appendix by N. Byott and L. Vendramin)
• Non-existence of Hopf-Galois structures and bijective crossed homomorphisms
• Hopf-Galois structures on a Galois $S_n$-extension

Cited by in corpus (4)

• Rota-Baxter groups, skew left braces, and the Yang-Baxter equation
• Hopf-Galois structures on finite extensions with almost simple Galois group
• Hopf-Galois structures on finite extensions with quasisimple Galois group
• Non-abelian simple groups which occur as the type of a Hopf-Galois structure on a solvable extension