Normal closure and injective normalizer of a group homomorphism

2014/03/14 by Farjoun, Emmanuel D., Segev, Yoav
#18A40 #20F28 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #Primary: 20E22 #Secondary: 20J06

paper · doi:10.48550/arxiv.1403.3501

Abstract

Let φ\colonΓ→ G be a homomorphism of groups. We consider factorizations Γ\xrightarrowf M\xrightarrowg G of φ 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.

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