Table of Contents
Define Formal Group
A formal group is a mathematical structure consisting of a set of elements together with a binary operation that associates two elements of the set to form a new element. The binary operation must satisfy four properties: closure, associativity, identity, and inverses.
What are Binary Operations?
The term binary operation refers to an arithmetic or logical operation that takes two operands. The most common binary operations are addition (+), subtraction (-), multiplication (*), and division (/).
Introduction to Groups
Groups are collections of people who share a common interest, activity, or purpose. Group members may interact with each other to achieve a common goal, or they may simply share a common bond.
There are many different types of groups, including social groups, work groups, and interest groups. Social groups are typically made up of people who are friends or acquaintances. Work groups are made up of people who are employees of the same company or organization. Interest groups are made up of people who share a common interest, such as environmentalism or gun rights.
Group members typically have some level of communication with each other. This communication can be direct, such as in a meeting, or indirect, such as through a social media site. Group members may also collaborate on tasks or projects.
Groups can be beneficial to their members because they provide a sense of belonging, networking opportunities, and social support. Groups can also be harmful to their members if the group is abusive or if the members are not able to meet the group’s expectations.
Formal Definition of a Group
A group is a set of elements together with a binary operation, usually denoted by , that satisfies the following four properties:
Closure: For all a, b in the group, a · b is also in the group.
Associativity: For all a, b, and c in the group, a · (b · c) = (a · b) · c.
Identity Element: There exists an element e in the group such that for all a in the group, a · e = e · a = a.
Inverses: For all a in the group, there exists an element a-1 in the group such that a · a-1 = e, where e is the identity element.
