Engineering↗
Cohomology of Finite Cyclic Groups
The periodic free resolution for a finite cyclic group, the resulting periodic cohomology, norm and difference maps, and Tate cohomology.
Tag
Cohomology of Groups
Every page in the KEVOS library tagged Cohomology of Groups. 11 pages.
Engineering↗
Cohomology of Products and Coproducts of Groups
Cohomology of a direct product via Kunneth, cohomology of a free product as a direct sum, Mayer-Vietoris for amalgamated products, and the contrast between the two constructions.
Engineering↗
Definition of Group Homology and Cohomology
Group homology and cohomology as derived functors of coinvariants and invariants, equivalently as Tor and Ext over the group ring, with the basic properties and long exact seque…
Engineering↗
Derivations and the Semidirect Product
Derivations and principal derivations, the semidirect product, complements and their conjugacy, and the interpretation of H^1 as classifying complements.
Engineering↗
Group Extensions and H2
Group extensions with abelian kernel, factor sets and their equivalence, the classification by H^2, central extensions, and the obstruction interpretation.
Engineering↗
H2, Hopf's Formula and the Schur Multiplier
Hopf's formula for H_2 in terms of a free presentation, the Schur multiplier, universal central extensions, and the connection to the lower central series.
Engineering↗
Low-Dimensional Group Cohomology
Explicit descriptions of H^0, H^1 and H_1 for group cohomology, with trivial and non-trivial coefficients, and their interpretations.
Engineering↗
Resolutions for Group Cohomology
The bar resolution and its normalised version, cocycles and coboundaries in explicit terms, resolutions from presentations, and the trade-off between explicitness and size.
Engineering↗
Subgroups: Restriction, Corestriction and Transfer
Restriction and corestriction (transfer) maps, the index formula, consequences for torsion, and detection on Sylow subgroups.
Engineering↗
The Five-Term Exact Sequence
The five-term exact sequence for a normal subgroup, inflation and restriction maps, transgression, and its derivation from the Lyndon-Hochschild-Serre spectral sequence.
Engineering↗
The Group Ring and the Augmentation Ideal
The group ring, modules over it as representations, the augmentation map and its kernel, and the relation between the augmentation ideal and the abelianisation.
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