TY - CHAP

T1 - Relative Schur multipliers and universal extensions of group homomorphisms

AU - Farjoun, Emmanuel D.

AU - Segev, Yoav

N1 - Publisher Copyright:
© 2017 American Mathematical Society.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We discuss an extension to a relative case of the construction by Schur of the universal central extension of a perfect group G, with the kernel being the Schur multiplier group H2(G; ℤ). In this note, starting with any group homomorphism f : Γ → G, which is surjective upon abelianization, we construct a universal central extension u: U ↠ G, relative to f with the same surjective property, such that for any central extension m: M ↠ G, relative to f, there is a unique homomorphism U → M with the obvious commutation condition. The kernel of u is the relative Schur multiplier group H2(G, Γ Z) as below. The case where G is perfect corresponds to Γ = 1. Upon repetition, for finite groups, this also gives a universal hypercentral factorization of the map f : Γ → G. We observe that our construction of those relative universal central extensions yield homological obstructions to lifting solutions of equations in a perfect group G to its Schur universal central extension E ↠ G.

AB - We discuss an extension to a relative case of the construction by Schur of the universal central extension of a perfect group G, with the kernel being the Schur multiplier group H2(G; ℤ). In this note, starting with any group homomorphism f : Γ → G, which is surjective upon abelianization, we construct a universal central extension u: U ↠ G, relative to f with the same surjective property, such that for any central extension m: M ↠ G, relative to f, there is a unique homomorphism U → M with the obvious commutation condition. The kernel of u is the relative Schur multiplier group H2(G, Γ Z) as below. The case where G is perfect corresponds to Γ = 1. Upon repetition, for finite groups, this also gives a universal hypercentral factorization of the map f : Γ → G. We observe that our construction of those relative universal central extensions yield homological obstructions to lifting solutions of equations in a perfect group G to its Schur universal central extension E ↠ G.

KW - Central extension

KW - Hypercenter

KW - Relative schur multiplier

KW - Second homology

KW - Universal factorization

UR - http://www.scopus.com/inward/record.url?scp=85029380896&partnerID=8YFLogxK

U2 - 10.1090/conm/682/13805

DO - 10.1090/conm/682/13805

M3 - Chapter

AN - SCOPUS:85029380896

T3 - Contemporary Mathematics

SP - 65

EP - 80

BT - Contemporary Mathematics

PB - American Mathematical Society

ER -