Precrossed modules and Galois theory

Central extensions of precrossed and crossed modules

The notion of centrality for crossed modules was introduced by Norrie in her thesis [7], in which she studied the category of crossed modules CM from an algebraic point of view, showing suitable generalizations of group theoretic concepts and results. Subsequently, Norrie’s approach was followed by Carrasco, Cegarra and R.-Grandjeán. In [5] they proved that CM is an algebraic category (i.e. the...

Galois modules and class field theory Boas

10. Galois modules and class field theory Boas Erez In this section we shall try to present the reader with a sample of several significant instances where, on the way to proving results in Galois module theory, one is lead to use class field theory. Conversely, some contributions of Galois module theory to class fields theory are hinted at. We shall also single out some problems that in our op...

More about Homological Properties of Precrossed Modules

Homology groups modulo q of a precrossed P-module in any dimensions are deened in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second ho-mology group modulo q of a precrossed P-module which shows that for q = 0 our deenition is a natural extension of Conduch e and El-lis' deenition CE]. Some other properties of homologies of precross...

GALOIS p-GROUPS AND GALOIS MODULES

The smallest non-abelian p-groups play a fundamental role in the theory of Galois p-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups — as well as other closely related, larger p-groups — occur as Galois groups over given base fields. We show further how the appearance of some Galois...

On universal central extensions of precrossed and crossed modules