ABELIAN GROUP Meaning and
Definition
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An Abelian group is a mathematical structure in abstract algebra, which consists of a set equipped with a binary operation that satisfies a specific set of properties. More formally, an Abelian group is a set G along with a binary operation, usually denoted as +, that satisfies the following conditions:
1. Closure: For any two elements a and b in G, the result of the operation a + b is also an element of G.
2. Associativity: For any three elements a, b, and c in G, the operation is associative, meaning that (a + b) + c = a + (b + c).
3. Identity element: There exists an identity element, usually denoted as 0 or e, such that for any element a in G, a + 0 = 0 + a = a.
4. Inverse elements: For every element a in G, there exists an inverse element -a such that a + (-a) = (-a) + a = 0.
5. Commutativity: The operation is commutative, meaning that for any two elements a and b in G, a + b = b + a.
The name "Abelian group" is derived from the Norwegian mathematician Niels Henrik Abel, who made significant contributions to the study of these groups. Abelian groups are named as such because they obey the commutative property, which was first investigated by the mathematician Évariste Galois and later generalized by Abel.
Abelian groups serve as important examples in mathematics and have numerous applications in various areas such as number theory, algebraic geometry, and physics. Common examples of Abelian groups include the group of integers under addition, the group of rational numbers under addition, and the group of real numbers
Common Misspellings for ABELIAN GROUP
- zbelian group
- sbelian group
- wbelian group
- qbelian group
- avelian group
- anelian group
- ahelian group
- agelian group
- abwlian group
- abslian group
- abdlian group
- abrlian group
- ab4lian group
- ab3lian group
- abekian group
- abepian group
- abeoian group
- abeluan group
- abeljan group
Etymology of ABELIAN GROUP
The term "Abelian group" is named after the Norwegian mathematician Niels Henrik Abel.
Abel was born in 1802 and made significant contributions to several areas of mathematics, including group theory. He is best known for his work on the unsolvability of quintic equations, but his investigations also led to the development of the concept of an Abelian group.
The term was introduced by the German mathematician Ferdinand Georg Frobenius in 1879 to honor Abel's achievements in group theory. An Abelian group is a mathematical structure with a set of elements and an associative binary operation that satisfies the commutative property, meaning that the order of the elements does not affect the result of the operation.
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