Tag
Cohomology of Groups
11 resources tagged “Cohomology of Groups” across the knowledge library.
11 items
Guide6 Aug 2026Cohomology of Finite Cyclic GroupsThe periodic free resolution for a finite cyclic group, the resulting periodic cohomology, norm and difference maps, and Tate cohomology.3 min readRead MoreGuide6 Aug 2026Cohomology of Products and Coproducts of GroupsCohomology 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.3 min readRead MoreGuide6 Aug 2026Definition of Group Homology and CohomologyGroup 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 sequences.3 min readRead MoreGuide6 Aug 2026Derivations and the Semidirect ProductDerivations and principal derivations, the semidirect product, complements and their conjugacy, and the interpretation of H^1 as classifying complements.3 min readRead MoreGuide6 Aug 2026Group Extensions and H2Group extensions with abelian kernel, factor sets and their equivalence, the classification by H^2, central extensions, and the obstruction interpretation.4 min readRead MoreGuide6 Aug 2026H2, Hopf's Formula and the Schur MultiplierHopf'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.3 min readRead MoreGuide6 Aug 2026Low-Dimensional Group CohomologyExplicit descriptions of H^0, H^1 and H_1 for group cohomology, with trivial and non-trivial coefficients, and their interpretations.3 min readRead MoreGuide6 Aug 2026Resolutions for Group CohomologyThe bar resolution and its normalised version, cocycles and coboundaries in explicit terms, resolutions from presentations, and the trade-off between explicitness and size.3 min readRead MoreGuide6 Aug 2026Subgroups: Restriction, Corestriction and TransferRestriction and corestriction (transfer) maps, the index formula, consequences for torsion, and detection on Sylow subgroups.3 min readRead MoreGuide6 Aug 2026The Five-Term Exact SequenceThe five-term exact sequence for a normal subgroup, inflation and restriction maps, transgression, and its derivation from the Lyndon-Hochschild-Serre spectral sequence.3 min readRead MoreGuide6 Aug 2026The Group Ring and the Augmentation IdealThe group ring, modules over it as representations, the augmentation map and its kernel, and the relation between the augmentation ideal and the abelianisation.3 min readRead More