FINITELY PRESENTED GROUP Meaning and
Definition
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A finitely presented group is a mathematical concept that refers to a group defined by a finite set of generators and a finite set of relations among these generators. In other words, it is a group that can be defined using a finite list of elements that generate the group, along with a finite set of equations or inequalities that define how these elements relate to each other.
More formally, a finitely presented group G can be defined as G = <S | R>, where S is a finite set of generators and R is a finite set of relations. The generators are the basic building blocks that generate all elements of the group, while the relations specify how these elements interact with each other.
The relations can be expressed as words formed by the generators and their inverses. These words represent equations or inequalities that must hold true within the group. By manipulating these relations, one can obtain the group elements and explore the group's properties.
Finitely presented groups are essential in the study of group theory, algebraic topology, and geometric group theory. They provide a way to precisely describe and classify many different types of groups, enabling mathematicians to analyze their structure and properties. They are also used in a wide range of applications, including cryptography, physics, and computer science.
