Hopf–Galois structures on finite extensions with quasisimple Galois group

Abstract

Let L/K be a finite Galois extension of fields with Galois group G. It is known that L/K admits exactly two Hopf–Galois structures when G is non‐abelian simple. In this paper, we extend this result to the case when G is quasisimple.

Non‐abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension

“…In fact, other than the obvious examples $$\rho (G)$$ and $$\lambda (G)$$, there are no other regular subgroups of $$\mathrm{Hol}(G)$$. In the language of Hopf–Galois structures, this means that (a)the only Hopf–Galois structures on a Galois $$G$$‐extension are the classical and canonical nonclassical ones (in the sense of [15]). In the language of skew braces, this means that (b)the only group operations $$\circ$$ on $$G = (G,+)$$ for which $$(G,+,\circ )$$ is a skew brace are the trivial and almost trivial ones, given by $$x\,\circ\, y = x+y$$ and $$x\,\circ\, y= y+x$$, respectively. The same is true when $$G$$ is fixed to be a finite quasi‐simple group [20].…”

“…Conjecture 1.1 is known to be true in some special cases; see [4,[18][19][20] for examples. Let us also remark that Byott has made some significant progress regarding this conjecture in the preprint arXiv:2205.13464.…”