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On the Cohomological Dimension of Soluble Groups

Published online by Cambridge University Press:  20 November 2018

D. Gildenhuys  and
R. Strebel
D. Gildenhuys
Affiliation:
Department of Mathematics, McGill UniversityBurnside HallMontreal, PQ, Canada H3A 2K6
R. Strebel
Affiliation:
Department of Mathematics, University of IllinoisUrbana, Illinois 61801
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Abstract

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It is known that every torsion-free soluble group G of finite Hirsch number hG is countable, and its homological and cohomological dimensions over the integers and rationals satisfy the inequalities

We prove that G must be finitely generated if the equality hG = cdQG holds. Moreover, we show that if G is a countable soluble group of finite Hirsch number, but not necessarily torsion-free, and if hG = cdQG, then hḠ = cdQḠ for every homomorphic image Ḡ of G.

Keywords

00140015soluble groupsHirsch numbershomological and cohomological dimension
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 24 , Issue 4 , 01 December 1981 , pp. 385 - 392
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Baer, R. and Heineken, H., Radical groups of finite abelian subgroup rank. Illinois J. Math. 16 (1972), 533-580.Google Scholar
2. Bieri, R., Über die cohomologische Dimension der auflösbaren Gruppen. Math. Z. 128 (1972), 235-243.Google Scholar
3. Bieri, R. and Strebel, R., Almost finitely presented soluble groups. Comment. Math. Helv. 53 (1978), 258-278.Google Scholar
4. Bieri, R., Homological dimension of discrete groups. Queen Mary College Lecture Notes, QMC (University of London) 1976.Google Scholar
5. Cohen, D. E., Groups of cohomological dimension one. Lecture Notes in Maths. Springer- Verlag. Berlin-Heidelberg-New York, 1972.Google Scholar
6. Čarin, V. S., On soluble groups of type A4. Mat. Sb. 52 (94) (1960), 895-914.Google Scholar
7. Gildenhuys, D., Classification of soluble groups of cohomological dimension two. Math. Z. 166, (1979), 21-25.Google Scholar
8. Gruenberg, K. W., Cohomological topics in group theory. Springer Lecture Notes in Mathematics 143. Springer-Verlag, Berlin-Heidelberg-New York, 1970.Google Scholar
9. Hall, P., On the finiteness of certain soluble groups. Proc. London Math. Soc. (3) 9 (1959), 592-622.Google Scholar
10. Jensen, C. U., Les fondeurs derives de et leurs applications en théorie des modules. Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 1972.Google Scholar
11. Robinson, D. J. S., On the cohomology of soluble groups of finite rank. J. Pure and Applied Alg. 6 (1975), 155-164.Google Scholar
12. Stammbach, U., On the weak homological dimension of the group algebra of soluble groups. J. London Math. Soc. (2) 2 (1970), 657-570.Google Scholar
13. Strebel, R., On homological duality. J. Pure and applied Alg. 8 (1976), 75-96.Google Scholar