Hopf-Galois structures on non-normal field extensions of squarefree degree

Greither and Pareigis [3] showed that a finite separable extension of
fields L=KL=K may admit many Hopf-Galois structures, and that finding
them all can be reduced to a problem in group theory. When L=K is
also normal (so that it is a Galois extension in the classical sense), the
Hopf-Galois structures have been determined in a number of special
cases. For example, Alabdali and Byott were able to determine all
Hopf-Galois structures on an arbitrary Galois extension of squarefree
degree [1, 2], using the classification of the groups of squarefree order.
The aim of this project is to investigate how far the results of [2] can
be generalised to non-normal extensions of squarefree degree. Instead
of groups of squarefree order, one now needs to consider (transitive)
permutation groups of squarefree degree, and no classification of these
is available. However, not all such permutation groups can arise from
Hopf-Galois structures, since the groups of interest must occur as
transitive subgroups of the holomorph of a group of squarefree order.
The difficulty is then to decide when subgroups of the holomorphs
of two different groups of squarefree order n give rise to Hopf-Galois
structures on the same degree n field extension. Initial work by Byott
and Martin-Lyons (undertaken with a London Mathematical Society
Undergraduate Research Bursary, and not yet written up) indicates
that this can be done in the special case n = pq where p, q are primes
and p = 2q + 1 (so p is a Sophie Germain prime). This PhD project
will seek to apply to the same approach in other cases.

References
[1] A.Alabdali and N.P.Byott: Hopf-Galois structures on cyclic field
extensions of squarefree degree. J. Algebra 493 (2018), 1-19.
[2] A.Alabdali and N.P.Byott: Hopf-Galois structures of squarefree degree,
J. Algebra 559 (2020), 58-86
[3] C.Greither and B.Pareigis: Hopf Galois theory for separable field extensions.
J. Algebra 106 (1987), no. 1, 239-258.