Abelian group
Definition and Examples
An Abelian group, named after the Norwegian mathematician Niels Henrik Abel, is a set equipped with an associative binary operation that has an identity element and every element in the set has an inverse. In more concrete terms, an Abelian group is a mathematical structure that allows for the addition (or multiplication) of elements in a way that is commutative, meaning the order of the elements does not affect the result, and there exists an element that when added to any other element does not change the latter.
For instance, the set of integers under addition forms an Abelian group. The operation is addition, the identity element is zero (since adding zero to any integer does not change the value), and the inverse of any integer n is its negation -n (since adding n and -n yields zero, the identity element). This group is denoted as (Z, +).
Properties
Abelian groups have several important properties that make them a fundamental object of study in abstract algebra. These properties include:
- Closure: If a and b are elements in the group, then the result of the operation a * b is also in the group.
- Associativity: For any elements a, b, and c in the group, (a * b) * c equals a * (b * c).
- Identity element: There exists an element e in the group such that for every element a in the group, the equations e * a and a * e return a.
- Inverse element: For each element a in the group, there exists an element b in the group such that a * b and b * a both equal the identity element.
- Commutativity: For all a and b in the group, a * b equals b * a.
Subgroups
A subgroup of an Abelian group is a subset of the group that is itself an Abelian group under the operation of the larger group. For example, the set of even integers is a subgroup of the group of all integers (under addition), because the sum of any two even integers is still an even integer.
Homomorphisms and Isomorphisms
In the context of Abelian groups, a homomorphism is a function between two Abelian groups that preserves the group operation. More formally, if (G, *) and (H, ·) are Abelian groups, a function f: G → H is a homomorphism if for all a and b in G, it holds that f(a * b) = f(a) · f(b).
An isomorphism is a bijective (one-to-one and onto) homomorphism. If there is an isomorphism between two Abelian groups, they are said to be isomorphic, and they are essentially the same group in a structural sense.
Applications
Abelian groups have wide applications in various fields of mathematics, including number theory, algebraic geometry, and topology. For instance, they are used in the definition of cohomology, which is a powerful tool in algebraic topology.