Definition

Noun: A commutative group is a mathematical structure consisting of a set of elements together with an operation (like addition or multiplication) that combines any two elements to form a third. For this structure to be a commutative group, it must satisfy four specific axioms: closure, associativity, identity, and invertibility. Crucially, it must also satisfy the commutative law, meaning the order in which you combine two elements does not change the result. This property is also known as Abelian, named after mathematician Niels Henrik Abel.

Usage Examples

• The set of integers under addition, (ℤ, +), is the most familiar example of a commutative group.
• In linear algebra, the set of all n-dimensional vectors with real-number components forms a commutative group under vector addition.
• The structure (ℝ*, ·) of non-zero real numbers under multiplication is also a commutative group.

Advanced Usage

• In abstract algebra, the term Abelian group is used synonymously with commutative group. The study of these groups is often simpler than that of non-commutative groups.
• The property that defines a commutative group can be stated formally: for all elements and in the group, ∘ = ∘ , where "∘" denotes the group operation.

Variants and Related Words

• Abelian group (n): A direct synonym for a commutative group.
• Group (n): The more general algebraic structure which may or may not be commutative.
• Non-commutative group (n): A group whose operation does not satisfy the commutative law (e.g., the group of invertible matrices under matrix multiplication).

Synonyms

• Abelian group

Related Concepts (Not Phrasal Verbs/Idioms)

• Commutative Law/Property: The specific axiom (a ∘ b = b ∘ a) that distinguishes a commutative group from a general group.
• Ring: An algebraic structure that has two operations, where the set under one of the operations (usually addition) is required to form a commutative group.

Noun

1. a group that satisfies the commutative law