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

Algebra Seminar

21st October 2020, 2:30 pm – 3:30 pm

Online, Zoom

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.

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