FREE GROUP Meaning and
Definition
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A free group is a fundamental concept in algebraic group theory. It is defined as a group that is freely generated by a given set of elements, subject to the condition that no relation between these elements holds, except for the group's basic properties.
More precisely, let S be a set of elements. Then, a free group on S, denoted as F(S), is a group that can be generated by the elements of S, where each element of S is considered as an individual generator. In F(S), there are no additional relations imposed on these generators, apart from the usual group axioms such as closure, associativity, and the existence of inverses. This means that the elements of S and their inverses are the only elements that appear in F(S).
The concept of a free group is often used to study other groups by considering the homomorphisms between them. Since any group can be viewed as a quotient of some free group, understanding the properties and structure of free groups provides insights into the properties and structure of other groups.
One important aspect of free groups is that they are non-abelian, meaning that their operation is not commutative in general. This non-commutativity allows for the existence of non-trivial relations and diverse algebraic structures within the group.
Common Misspellings for FREE GROUP
- dree group
- cree group
- vree group
- gree group
- tree group
- rree group
- feee group
- fdee group
- ffee group
- ftee group
- f5ee group
- f4ee group
- frwe group
- frse group
- frde group
- frre group
- fr4e group
- fr3e group
- frew group
- fres group
Etymology of FREE GROUP
The term "free group" originated from mathematics, specifically in the field of group theory. The word "free" is used in a technical sense, indicating that the group is free in the sense that it has the minimum number of relations among its elements.
The term "group" itself comes from the late 19th century, derived from the German word "Gruppe", which means a collection or set. In mathematics, a group is a fundamental algebraic structure comprising a set of elements along with an operation that satisfies certain properties, such as closure, associativity, identity, and invertibility.
The concept of a free group was introduced in the early 20th century by a German mathematician named Max Dehn. He used the term "free group" to describe a group that can be generated by a set of elements without any specific relations between them.
