Lecture

Orbit-Stabilizer Theorem: Conjugacy Classes and Direct Products

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In course

MATH-310: Algebra

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

Description

This lecture covers the Orbit-Stabilizer theorem, conjugacy classes, and centralizers in finite groups. It explains the class equation, conjugacy classes in symmetric groups, and examples in S_4. The lecture also discusses simple groups, simple abelian finite groups, and the concept of direct product of groups.

Instructor

Anna Lachowska

Teaching & PhD Courses Algebra MATH-310 This is an introduction to modern algebra: groups, rings and fields. Analysis I MATH-101(e) We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable. Analysis II MATH-106(e) The course studies fundamental concepts of analysis and the calculus of functions of several variables. Representation theory I - finite groups MATH-314 This is a standard c

Official source

Ontological neighbourhood

Mathematics

Algebra: Representation theory, Abstract algebra

Related concepts (32)

Classification of finite simple groups

In mathematics, the classification of finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

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.

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.

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.

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.

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