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A note on central group extensions

Published online by Cambridge University Press:  09 April 2009

G. J. Hauptfleisch
G. J. Hauptfleisch
Affiliation:
Tulane University New Orleans Louisiana, U.S.A. and Rand Afrikaans UniversityJohannesburg Republic of South Africa
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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.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 4 , June 1973 , pp. 428 - 429
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Hauptfleisch, G.J., ‘Quasi-group extensions of Abelian groups’ (Thesis, Leiden, 1965).Google Scholar