With the advent of the recent determination of high-resolution crystal structures of bovine rhodopsin and human beta2 adrenergic receptor (beta2AR), there are still many structure-function relationships to be learned from other G protein-coupled receptors (GPCRs). Many of the pharmaceutically interesting GPCRs cannot be modeled because of their amino acid sequence divergence from bovine rhodopsin and beta2AR. Structure determination of GPCRs can provide new avenues for engineering drugs with greater potency and higher specificity. Several obstacles need to be overcome before membrane protein structural biology becomes routine: over-expression, solubilization, and purification of milligram quantities of active and stable GPCRs. Coordinated iterative efforts are required to generate any significant GPCR over-expression. To formulate guidelines for GPCR purification efforts, we review published conditions for solubilization and purification using detergents and additives. A discussion of sample preparation of GPCRs in detergent phase, bicelles, nanodiscs, or low-density lipoproteins is presented in the context of potential structural biology applications. In addition, a review of the solubilization and purification of successfully crystallized bovine rhodopsin and beta2AR highlights tools that can be used for other GPCRs.

By work of C. Greither and B. Pareigis as well as N. P. Byott, the enumeration of Hopf-Galois structures on a Galois extension of fields with Galois group G may be reduced to that of regular subgroups of Hol(N ) isomorphic to G as N ranges over all groups of order |G|, where Hol(−) denotes the holomorph. In this paper, we shall give a description of such subgroups of Hol(N ) in terms of bijective crossed homomorphisms G −→ N , and then use it to study two questions related to non-existence of Hopf-Galois structures.

We exhibit infinitely many natural numbers [Formula: see text] for which there exists at least one insolvable group of order [Formula: see text], and yet the holomorph of every solvable group of order [Formula: see text] has no insolvable regular subgroup. We also solve Problem 19.90(d) in the Kourovka notebook.

Abstract. Let K be a number field with ring of integers O K and G a finite group of odd order. If K h is a weakly ramified G-Galois K-algebra, then its square root A h of the inverse different is a locally free O K G-module and hence determines a class in the locally free class group Cl(O K G) of O K G. We show that for G abelian and under suitable assumptions, the set of all such classes is a subgroup of Cl(O K G).

Let L/K be a finite Galois extension whose Galois group G is non-abelian and characteristically simple. Using tools from graph theory, we shall give a closed formula for the total number of Hopf-Galois structures on L/K with associated group isomorphic to G.