In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules is a motivation, but the cohomology can be defined using various constructions. There is a dual theory, group homology, and a generalization to non-abelian coefficients.
These algebraic ideas are closely related to topological ideas. The group cohomology of a group G can be thought of as, and is motivated by, the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of can be thought of as the singular cohomology of the circle, and similarly for and .
A great deal is known about the cohomology of groups, including interpretations of low dimensional cohomology, functorality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.
Read more about Group Cohomology: Motivation, Formal Constructions, Non-abelian Group Cohomology, Connections With Topological Cohomology Theories, History and Relation To Other Fields
Famous quotes containing the word group:
“It’s important to remember that feminism is no longer a group of organizations or leaders. It’s the expectations that parents have for their daughters, and their sons, too. It’s the way we talk about and treat one another. It’s who makes the money and who makes the compromises and who makes the dinner. It’s a state of mind. It’s the way we live now.”
—Anna Quindlen (20th century)