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Cohomology Theory for Non-Normal Subgroups and Non-Normal Fields*

Published online by Cambridge University Press:  18 May 2009

Iain T. Adamson
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Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:

choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that if

is an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequence

may be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequence

is exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 2 , Issue 2 , October 1954 , pp. 66 - 76
Copyright
Copyright © Glasgow Mathematical Journal Trust 1954

References

REFERENCES

(1)Artin, E., Algebraic Numbers and Algebraic Functions (New York, 1951).Google Scholar
(2)Artin, E. and Tate, J. T., Class Field Theory, Princeton seminar notes, 1951–1952, (to appear).Google Scholar
(3)Brauer, R., “On Splitting Fields of Simple Algebras”, Annals of Math., vol. 48 (1947), pp. 79–90.CrossRefGoogle Scholar
(4)Eilenberg, S., “Homology of Spaces with Operators, I”, Trans. Amer. Math. Soc., vol. 61 (1947), pp. 378–417.CrossRefGoogle Scholar
(5)Hochschild, G. and Nakayama, T., “Cohomology in Class Field Theory”, Annals of Math., vol. 55 (1952), pp. 348–66.CrossRefGoogle Scholar
(6)Tate, J. T., “The Higher Dimension Cohomology Groups of Class Field Theory”, Annals of Math., vol. 56 (1952), pp. 294–7.CrossRefGoogle Scholar