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Lecture

Algebra: Group Actions and Direct Products

In course

MATH-310: Algebra

This is an introduction to modern algebra: groups, rings and fields. ...

Description

This lecture covers the action of a group on itself by conjugations, conjugacy classes, centralizers, and the class equation of a finite group. It also discusses the direct product of groups, including basic properties and the classification of finite abelian groups.

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Instructor

Anna Lachowska

Official source

Ontological neighbourhood

Mathematics

Algebra: Representation theory, Abstract algebra

Related concepts (32)

Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

Conjugacy class

In mathematics, especially group theory, two elements and of a group are conjugate if there is an element in the group such that This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under for all elements in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure.

Finitely generated abelian group

In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate . Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers, , are a finitely generated abelian group. The integers modulo , , are a finite (hence finitely generated) abelian group.

Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.

Direct product

In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the , which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.

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Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Classification of Finite Abelian Groups

Covers the classification of finite abelian groups and the unique sequence of positive integers for isomorphism.

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Explores chirality, group completeness, Abelian groups, conjugate classes, and isomorphic groups in symmetry and group theory.

Direct Sums: Lemme 1.2

Delves into direct sums of abelian groups, showcasing their coproduct properties and universal verification.

Existence: Proof of Classification Theorem