FINITE GROUPS Meaning and
Definition
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Finite groups are mathematical structures consisting of a set of elements along with an operation that combines two elements and produces another element within the set. A finite group is distinguished by having a limited or finite number of elements in its set.
In more detail, a finite group G is defined as an ordered pair (S, *) where S is a non-empty set and * is a binary operation such that the following conditions (known as group axioms) hold true:
1. Closure: For any two elements a, b in S, the result of the operation a * b is also an element of S. In other words, the group is closed under the operation *.
2. Associativity: For any three elements a, b, c in S, the operation is associative, meaning (a * b) * c = a * (b * c).
3. Identity: There exists an element e in S, called the identity element, such that for any element a in S, a * e = e * a = a.
4. Inverse: For every element a in S, there exists an element b in S, called the inverse of a, such that a * b = b * a = e, where e is the identity element.
5. Finite set: The set S has a finite number of elements.
Finite groups are fundamental in various areas of mathematics and have important applications in fields such as physics and cryptography. They can be studied for their structural properties, symmetries, and representation theory, providing insights into the fundamental nature of mathematical objects and their relationships.
Etymology of FINITE GROUPS
The term "finite groups" consists of two separate words: "finite" and "groups".
The word "finite" comes from the Latin word "finis", meaning "end" or "boundary". It indicates that something has a limited or bounded nature and does not extend indefinitely.
The word "group" has its roots in Middle English and Old French, derived from the Latin word "gruppus" or "gruppo", meaning "a cluster" or "a bunch". It refers to a collection or a set of elements or individuals that exhibit a particular relationship or property.
When combined, the term "finite groups" denotes a collection or set of elements that is limited or bounded. In mathematics, a finite group is a group with a finite number of elements.
