We consider Hopf-Galois structures on separable (but not necessarily normal) field extensions $L/K$ of squarefree degree n. If $E/K$ is the normal closure of $L/K$ then $G=\mathrm{Gal}(E/K)$ can be viewed as a permutation group of degree n. We show that G has derived length at most 4, but that many permutation groups of squarefree degree and of derived length 2 cannot occur. We then investigate in detail the case where $n=pq$ where $q\ge 3$ and $p=2q+1$ are both prime. (Thus q is a Sophie Germain prime and p is a safeprime). We list the permutation groups G which can arise, and we enumerate the Hopf-Galois structures for each G. There are six such G for which the corresponding field extensions $L/K$ admit Hopf-Galois structures of both possible types.

我们考虑可分离（但不一定是正常）域扩展上的 Hopf-Galois 结构 $升/钾$的平方自由度n。如果$乙/钾$ 是正常关闭 $升/钾$ 然后 $G=\mathrm{加尔}(乙/钾)$可以看作是一个n 次的置换群。我们证明G 的导出长度至多为 4，但不能出现许多自由平方度和导出长度为 2 的置换群。然后我们详细调查这种情况$n=磷q$ 在哪里 $q\ge 3$ 和 $磷=2q+1$都是素数。（因此q是 Sophie Germain 素数，p是安全素数）。我们列出了可能出现的置换群G，并列举了每个G的 Hopf-Galois 结构。有六个这样的G对应的字段扩展$升/钾$ 承认两种可能类型的 Hopf-Galois 结构。