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Groups (Algebraic Structure) - Group Theory

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Groups (Algebraic Structure) - Group Theory
  • in abstract algebra, group theory studies the algebraic structures known as groups
  • a group is a set equipped with a binary operation
  • groups is an abstraction of symmetric-actions, as numbers are an abstraction of counts
  • 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 - Definition

A group (𝐺,·) is a set 𝐺 equipped with a binary operation · that satisfies the following properties shown below:

A binary operation on 𝐺 is a mapping 𝐺×𝐺 → 𝐺, that is, a correspondence that associates with each ordered pair of elements of 𝐺 a uniquely determined element of 𝐺.

Binary Operation Properties (𝑎, 𝑏, 𝑐 are arbitrary elements of the group 𝐺)

Description

Closed

Associativity

Identity

Invertibility

Commutativity

Binary Operation

is a monoid whose elements are invertible
is a loop whose binary operation is associative
is an inverse group with an identity element

Periodic Table of Finite Simple Groups


period-table-of-finite-simple-groups.webp

Group - Examples

Other

Resources