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 = 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 ...
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 = 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 ≥ 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.