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

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Abstract
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, -∞H1U, A')= 0 for all subgroups U of H, then a cohomology group sequence may be defined and is exact for -∞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.

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