5 citations · 17 across the 14 of their papers we have counts for
20 papers
Bidihedral skew braces
Alan Koch, Paul J. Truman
We classify skew braces with additive and multiplicative groups both isomorphic to the dihedral group . As a consequence, we obtain an alternative proof of the classificat…
Analogues of Sylow's first theorem, Cauchy's theorem, and Hall's theorem for skew braces
Paul J. Truman
We establish an unconditional analogue of Sylow's first theorem for finite skew braces, and deduce an analogue of Cauchy's theorem. We also prove an analogue of the existence part…
Bicyclic biskew braces
Hal Simpson, Paul J. Truman
We study finite braces whose additive and multiplicative groups are both cyclic. We reinterpret the classification of these braces from the perspective of regular subgroups of perm…
On some semidirect products of skew braces arising in Hopf-Galois theory
Paul J. Truman
We classify skew braces that are the semidirect product of an ideal and a left ideal. As a consequence, given a Galois extension of fields whose Galois group is the semidir…
Constructing skew bracoids via abelian maps, and solutions to the {Y}ang-{B}axter equation
Alan Koch, Paul J. Truman
We show how one can use the skew braces constructed using abelian maps to generate families of skew bracoids as defined by Martin-Lyons and Truman. Under certain circumstances, the…
Skew bracoids containing a skew brace
Ilaria Colazzo, Alan Koch, Isabel Martin-Lyons +1
Skew bracoids have been shown to have applications in Hopf-Galois theory. We show that a certain family of skew bracoids correspond bijectively with left cancellative semibraces. A…
Skew bracoids
Isabel Martin-Lyons, Paul J. Truman
Skew braces are intensively studied owing to their wide ranging connections and applications. We generalize the definition of a skew brace to give a new algebraic object, which we…
On -conjugate Hopf-Galois structures
Paul J. Truman
The Hopf-Galois structures admitted by a Galois extension of fields with Galois group correspond bijectively with certain subgroups of . We use a…
Abelian fixed point free endomorphisms and the Yang-Baxter equation
Alan Koch, Laura Stordy, Paul J. Truman
We obtain a simple family of solutions to the set-theoretic Yang-Baxter equation, one which depends only on considering special endomorphisms of a finite group. We show how such an…
Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions
Alan Koch, Paul J. Truman
Given a finite group , we study certain regular subgroups of the group of permutations of , which occur in the classification theories of two types of algebraic objects:…
Opposite skew left braces and applications
Alan Koch, Paul J. Truman
Given a skew left brace , we introduce the notion of an "opposite" skew left brace , which is closely related to the concept of the opposite of a group…
Hopf-Galois module structure of tamely ramified radical extensions of prime degree
Paul J Truman
Let be a number field and let be a tamely ramified radical extension of prime degree . If contains a primitive root of unity then is a c…
The structure of Hopf algebras giving Hopf-Galois structures on Quaternionic extensions
Stuart Taylor, Paul J Truman
Let be a Galois extension of fields with Galois group isomorphic to the quaternion group of order . We describe all of the Hopf-Galois structures admitted by , an…
Isomorphism problems for Hopf-Galois structures on separable field extensions
Alan Koch, Timothy Kohl, Paul J. Truman +1
Let be a finite separable extension of fields whose Galois closure has group . Greither and Pareigis have used Galois descent to show that a Hopf algebra givin…
Normality and Short Exact Sequences of Hopf-Galois Structures
Alan Koch, Timothy Kohl, Paul J. Truman +1
Every Hopf-Galois structure on a finite Galois extension where corresponds uniquely to a regular subgroup , normalized by $λ(G)\l…
The Structure of Hopf Algebras Acting on Dihedral Extensions
Alan Koch, Timothy Kohl, Paul J. Truman +1
We discuss isomorphism questions concerning the Hopf algebras that yield Hopf-Galois structures for a fixed separable field extension . We study in detail the case where …
Commutative Hopf-Galois module structure of tame extensions
Paul J. Truman
We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of -adic fields or number fields which is -…
Commuting Hopf-Galois Structures on a Separable Extension
Paul J. Truman
Let be a finite separable extension of local or global fields in any characteristic, let be two Hopf algebras giving Hopf-Galois structures on the extensio…
Canonical Nonclassical Hopf-Galois Module Structure of Nonabelian Galois Extensions
Paul J Truman
Let be a finite Galois extension of local or global fields in characteristic or with nonabelian Galois group , and let be a -stable fractional i…
Towards a Generalisation of Noether's Theorem to Nonclassical Hopf-Galois Structures
Paul J. Truman
We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of -adic fields and number fields which are at most tamely ramified. W…