scholarly journals A note on Abelian group extensions

1962 ◽  
Vol 12 (4) ◽  
pp. 1401-1403 ◽  
Author(s):  
Ronald Nunke
Keyword(s):  
Abelian Group ◽  
Group Extensions

scholarly journals A note on central group extensions

1973 ◽  
Vol 15 (4) ◽  
pp. 428-429 ◽  
Author(s):  
G. J. Hauptfleisch
Keyword(s):  
Abelian Group ◽  
Homomorphic Image ◽  
Free Group ◽  
Central Extension ◽  
Dihedral Group ◽  
Group Extension ◽  
Central Group ◽  
Group Extensions

If A, B, H, K are abelian group and φ: A → H and ψ: B → K are epimorphisms, then a given central group extension G of H by K is not necessarily a homomorphic image of a group extension of A by B. Take for instance A = Z(2), B = Z ⊕ Z, H = Z(2), K = V4 (Klein's fourgroup). Then the dihedral group D8 is a central extension of H by K but it is not a homomorphic image of Z ⊕ Z ⊕ Z(2), the only group extension of A by the free group B.

Representations of holomorphs of group extensions with Abelian kernels

1975 ◽  
Vol 78 (3) ◽  
pp. 357-368 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Finite Rank ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Group Extensions

This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.

scholarly journals Automorphisms of abelian group extensions

2010 ◽  
Vol 324 (4) ◽  
pp. 820-830 ◽  
Author(s):  
I.B.S. Passi ◽  
Mahender Singh ◽  
Manoj K. Yadav
Keyword(s):  
Abelian Group ◽  
Group Extensions

scholarly journals Alexander invariants and cohomology jump loci in group extensions

2021 ◽  
Author(s):  
Alexander I. Suciu
Keyword(s):  
Abelian Group ◽  
Group Extensions ◽  
Characteristic Varieties ◽  
Alexander Invariants ◽  
Split Extensions ◽  
Chen Ranks ◽  
Resonance Varieties

Abstract We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1→K→G→Q→1, where Q is an abelian group acting trivially on H1(K;ℤ), with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

Abelian group extensions and the axiom of constructibility

1979 ◽  
Vol 54 (1) ◽  
pp. 440-457 ◽  
Author(s):  
Paul C. Eklof ◽  
Martin Huber
Keyword(s):  
Abelian Group ◽  
Group Extensions

scholarly journals Ergodic properties of a class of discrete Abelian group extensions of rank-one transformations

2010 ◽  
Vol 119 (1) ◽  
pp. 1-22
Author(s):  
Chris Dodd ◽  
Phakawa Jeasakul ◽  
Anne Jirapattanakul ◽  
Daniel M. Kane ◽  
Becky Robinson ◽  
...  
Keyword(s):  
Abelian Group ◽  
Group Extensions ◽  
Ergodic Properties ◽  
Rank One ◽  
Discrete Abelian Group
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