Group cohomology

In mathematics, particularly in homological algebra, group cohomology is a set of tools used to study groups. It uses ideas from cohomology theory. This theory comes from algebraic topology and helps us understand groups better.

Group cohomology is closely tied to topology. For a simple group, its cohomology can be thought of as the singular cohomology of a special space. For example, the group cohomology of the integers Z can be viewed as the cohomology of a circle S¹. This connection lets mathematicians use topology to solve problems in group theory.

The study of group cohomology began in the 1920s and grew in the late 1940s. It is still an active area of research today. It has uses in abstract algebra, algebraic number theory, and many other fields.

Motivation

In group theory, we study groups by looking at their group representations. These representations can be expanded to things called G-modules. A G-module is a special kind of group M where each part of G acts in a certain way on M.

When we study G-modules, we can find special parts in M that stay the same no matter how G acts on them. These are called invariant elements. Group cohomology helps us understand how these invariant elements act when we compare different G-modules. It shows us how the invariants in one module relate to invariants in another. This gives us better understanding of the structure of the group G.

Definitions

Group cohomology is a part of mathematics that helps us study groups using ideas from shapes and spaces. Think of it as a way to understand groups by looking at how they act on other mathematical objects.

Imagine you have a group, and you want to see how it interacts with a special kind of structure called a G-module. Group cohomology gives us tools to explore these interactions. It connects group theory with algebraic topology, a branch of math that studies shapes and spaces.

One way to understand group cohomology is by using something called cochain complexes. These are like step-by-step lists of rules that help us calculate important properties of the group. By studying these rules, we can learn more about the group's structure and behavior.

Group homology

Group homology is a part of mathematics that studies groups using methods from algebra. It is closely related to group cohomology but looks at different properties. For a group G and a G-module M, group homology looks at special constructions involving M.

This idea helps mathematicians understand the structure of groups by looking at how they interact with other mathematical objects. It uses tools from algebraic topology to study these relationships in an organized way.

Low-dimensional cohomology groups

The zero-degree cohomology group shows the module of invariants. The first and second degree cohomologies have simple meanings. The first cohomology group looks at special maps between groups, called crossed homomorphisms.

The second cohomology group, when the group action is trivial, corresponds to certain ways of extending a group. For example, it can classify special algebraic structures called division algebras.

Basic examples

Group cohomology is a way to study groups using ideas from geometry and topology. It shows how a group works with other math objects to learn more about the group.

One simple example is a finite cyclic group. This group repeats after a certain number of steps. By looking at how this group works with numbers, we can find its cohomology groups. These groups are collections of numbers that tell us about the group's structure. Another example is free groups. These groups are made by combining individual elements without extra rules. We can also calculate their cohomology using methods from topology. This is done by comparing the group to a space made from circles.

Properties

Group cohomology is a way to study groups using ideas from shapes and spaces. It looks at how a group works on certain things to learn more about the group's structure. This helps mathematicians understand groups better by linking them to shapes.

One important idea in group cohomology is the "long exact sequence." This sequence shows how the properties of a group and its smaller parts are connected. It helps mathematicians find important values called cohomology groups, which tell us more about the group. This sequence is very useful for solving problems in math and shapes.

Further examples

Group cohomology helps us understand how groups relate to each other. This is true when one group controls another in a special way, called a semi-direct product. We use ideas from topology, the study of shapes and spaces, to look at groups differently.

By thinking of groups as spaces, we can use special sequences to learn more about how they are built. These sequences connect the larger group to the smaller groups inside it, showing us more about how the whole group works.

Cohomology of finite groups

The cohomology groups of finite groups have a special property. They are all torsion for levels one and above. This means that, under certain conditions, these groups become zero.

For example, when the order of the group does not divide the characteristics of the field, the cohomology groups vanish.

Tate cohomology combines both homology and cohomology of a finite group. It has useful properties like long exact sequences and product structures. Tate cohomology is especially important in class field theory. For cyclic groups, Tate cohomology shows a repeating pattern every two steps.

Applications

Group cohomology has many useful applications in mathematics. One important area is algebraic K-theory, where it helps us understand the structure of rings and their associated groups. In this context, group cohomology looks at how certain sequences of groups become stable, meaning their properties become predictable.

Another key use is in studying projective representations in quantum mechanics. These are special ways groups can act on spaces, often with phases or "twists." Group cohomology helps classify these representations by studying the math behind these phases.

Extensions

Group cohomology is a way to study groups using ideas from geometry and algebra. It helps us learn about groups by seeing how they act on other math objects.

When groups have a special structure called a topology, we can use continuous actions to learn more about them. This leads to areas like Galois cohomology, which is important in number theory. We can also study groups using non-abelian cohomology, where group actions are not commutative, making the theory more complex.

History and relation to other fields

Group cohomology started from ideas studied long before it got its name in 1943–45. One early idea came from a theorem by David Hilbert in 1897. This theorem was later linked to work in Galois theory. Other mathematicians looked at similar ideas in the early 1900s when they studied group extensions.

In the 1940s, several mathematicians found group cohomology on their own. They included Samuel Eilenberg and Saunders Mac Lane in the United States, Heinz Hopf and Beno Eckmann in Switzerland, Hans Freudenthal in the Netherlands, and Dmitry Faddeev in the Soviet Union. Their work helped people use group cohomology in many areas, like algebraic number theory and physics.