The regular-subgroup solubility conjecture for finite soluble groups

Let $N$ be a finite soluble group, and let $G$ be a regular subgroup of the holomorph

$$\operatorname{Hol}(N)$$

of $N$, meaning that $G$ acts regularly on the underlying set of $N$. Regular-subgroup solubility conjecture. The group $G$ is soluble. This conjecture arises in both Hopf–Galois theory and the theory of skew braces; the paper investigates the broader question of insoluble transitive subgroups with soluble point stabilisers and reports classification results for irreducible solutions, but the stated conjecture itself is not resolved here.