## Abstract

Let ϕ: Γ→. G be a homomorphism of groups. We consider factorizations Γ→fM→gG of ϕ having certain universal properties. First we continue the investigation (see [4]) of the case where g is a universal normal map (our term for a crossed module). Then we introduce and investigate a seemingly new dual case, where f is a universal normal map. These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup.Iteration of these universal factorizations yields certain towers associated with the map ϕ we prove stability results for these towers. In one of the cases we get a generalization of the stability of the automorphisms tower of a center-less group. The case where g is a universal normal map is closely related to hyper-central group extensions, Bousfield's localizations, and the relative Schur multiplier H_{2}(G, Γ)=H_{2}(BG∪_{Bϕ}Cone(BΓ)).Although our constructions here have strong ties to topological constructions, we take here a group theoretical point of view.

Original language | English |
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Pages (from-to) | 1010-1043 |

Number of pages | 34 |

Journal | Journal of Algebra |

Volume | 423 |

DOIs | |

State | Published - 1 Feb 2015 |

## Keywords

- Automorphisms tower
- Central extension
- Crossed module
- Normal map
- Relative Schur multiplier