Timothy Kohl

Multiple Holomorphs of Dihedral and Quaternionic Groups

The holomorph of a group G is Norm B (λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups D n and quaternionic (dicyclic) groups Q n for n ≥ 3.

Computation of several cyclotomic Swan subgroups

Let Cl(OK[G]) denote the locally free class group, that is the group of stable isomorphism classes of locally free OK[G]-modules, where OK is the ring of algebraic integers in the number field K and G is a finite group. We show how to compute the Swan subgroup, T(OK[G]), of Cl(OK[G]) when K = Q(ζp), ζp a primitive p-th root of unity, G = C2, where p is an odd (rational) prime so that hp+ = 1 and 2 is inert in K/Q. We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial ˚ we do the computations obtaining for several primes p a nontrivial divisor of Cl(Z[ζp]C2). These calculations give an alternative proof that the fields Q(ζp) for p=11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

Hopf–Galois structures arising from groups with unique subgroup of order p

For

Group Rings and Hopf-Galois Theory in Maple

\Gamma

A class of profinite Hopf-Galois extensions over Q

a group of order

Classification of the Hopf Galois Structures on Prime Power Radical Extensions

mp

Groups of order 4p, twisted wreath products and Hopf–Galois theory

for

Regular permutation groups of order mp and Hopf Galois structures

p

Isomorphism problems for Hopf–Galois structures on separable field extensions

prime where

Normality and Short Exact Sequences of Hopf-Galois Structures

gcd(p,m)=1