Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Quotients of Groups by Relations of Equivalence

In course

MATH-113: Algebra I - fundamental structures

Le but de ce cours est d'introduire et d'étudier les notions de base de l'algèbre abstraite. ...

Description

This lecture discusses the concept of quotients of groups by relations of equivalence, exploring the conditions under which the set G / R is well defined. It covers the properties of equivalence relations, group operations, and the definition of a group. The lecture also delves into examples illustrating the application of these concepts, such as the group operation in modular arithmetic. Furthermore, it examines the properties of groups under specific relations, including the existence of inverses and the neutral element. The instructor demonstrates how to determine if a given set forms a group under a defined operation, providing insights into the fundamental principles of group theory.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Zsolt Patakfalvi

See CV here .

Dimitri Stelio Wyss

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Group theory

In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

Group (mathematics)

In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.

Equivalence class

In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set and an equivalence relation on the of an element in denoted by is the set of elements which are equivalent to It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of by and is denoted by .

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00.

Related lectures (52)

Commutative Groups: Foundations for Cryptography

Covers commutative groups and their significance in cryptography.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Classification of Extensions

Explores the classification of extensions in group theory, emphasizing split extensions and semi-direct products.

Recap of Group Theory

Provides a recap of group theory, defining a group as a set with a multiplication operation.

Group Theory Basics