Normal closure and injective normalizer of a group homomorphism

Abstract

Let $\phi :\Gamma \to G$ be a homomorphism of groups. We consider factorizations $\Gamma \stackrel{f}{\to }M\stackrel{g}{\to }G$ of $\phi$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have strong ties to topological constructions.

arXiv comment30 pages, to appear in J. Algebra