Understanding Abelian Groups: An Introduction to the Commutative Property in Mathematics

Abelian Group

A group that is also commutative

An Abelian group is a mathematical structure that satisfies the axioms of a group and also satisfies the commutative property, which states that the order of operations does not affect the result. Specifically, an Abelian group is a set G together with a binary operation (usually denoted as +) that satisfies the following properties:

1. Closure: For any a, b in G, the element a + b is also in G.
2. Associativity: For any a, b, c in G, (a + b) + c = a + (b + c).
3. Identity Element: There exists an element e in G such that for any a in G, a + e = a.
4. Inverse Element: For any a in G, there exists an element -a in G such that a + (-a) = e.
5. Commutativity: For any a, b in G, a + b = b + a.

The most common example of an Abelian group is the set of integers under addition (+). This structure satisfies all of the axioms of a group and the commutative property, since a + b = b + a for any integers a and b.

Another example of an Abelian group is the set of n × n matrices over a field F, denoted by Mn(F). Under matrix addition, Mn(F) satisfies the axioms of a group and the commutative property since A + B = B + A for any matrices A and B in Mn(F).

Abelian groups are important in many areas of mathematics, including abstract algebra, number theory, and geometry. They are used to describe symmetries, rotations, and transformations of geometric objects, among other applications.

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