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1-Cohomology and Splitting of Group Extensions

Published online by Cambridge University Press:  20 November 2018

G. N. Pandya  and
R. D. Bercov
G. N. Pandya
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
R. D. Bercov
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1
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The object of this note is to give simpler proofs of a splitting theorem of Gaschütz [1] and a related theorem for groups with operators by using cross-homomorphisms (1-cocycles) instead of 2-cohomology.

We recall that a cross-homomorphism or 1-cocycle from a group E to an abelian normal subgroup N of E is a map f from E to N such that f(ab)=(f(a))bf(b) for all a, b ∈ E where superscript denotes conjugation. Cocycles f and h are equivalent if for some n ∈ N have h(e)=nef(e)n-1 for all e ∈ E.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 3 , 01 September 1976 , pp. 369 - 371
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Gaschütz, W., Zur Erweiterungstheorie endlicher Gruppen. J. Math. 190, 93–107 (1952).Google Scholar
2. Hofmann, K. H., Zerfallung topologischer Gruppen. Math. Z. 84, 16–37 (1964)Google Scholar
3. Hubbert, B., Endliche Gruppen I, Springer-Verlag (1967).Google Scholar