Andrea Caranti

Regular subgroups of the holomorph, and their connections to the Yang Baxter equation, Hopf Galois structures, radical rings, and (skew) braces

The abstract holomorph of a group G is the natural semidirect
product of G by its automorphism group. It is isomorphic to its
permutational version, that is, the normaliser of the image of the
right regular representation of G in the group of permutations on
the set G.

Work of Greither and Pareigis, and of Byott, shows that the
classification of the Hopf Galois structures of number theory is
equivalent to the classification of the regular subgroups of a
suitable holomorph. Here a permutation group is said to be regular if,
given any two points, there is precisely one element of the group
taking one to the other.

The study of these regular subgroups is in turn equivalent to the
study of certain novel algebraic structures called skew braces. Braces
are generalisations of radical rings, that had been introduced by Rump
as a device to exhibit solution to the set-theoretic Yang-Baxter
equation of Mathematical Physics. Later Guarnieri and Vendramin
generalised the idea of Rump, and defined skew braces.

We will be discussing these topics mainly from the point of view of
the regular subgroups of the holomorph, addressing in particular the
problem of finding all the groups that, in a certain natural sense,
have the same holomorph as a given group.