Definition

1. Noun:
   • A group that satisfies the commutative law: In mathematics, specifically in abstract algebra, an abelian group is a set equipped with a binary operation (like addition) that combines any two elements to form a third. The defining property is that the operation is commutative; the order in which you combine elements does not change the result.

Usage

• The term "abelian group" is used exclusively in mathematical contexts to describe a specific algebraic structure. It is a formal, technical term.
• It functions as a countable noun (e.g., "an abelian group," "two abelian groups").

Examples

• Noun:
  • The set of integers under addition forms a classic example of an abelian group.
  • Every cyclic group is an abelian group.
  • The proof relies on the fact that the underlying structure is an abelian group.

Advanced Usage

• "Finitely generated abelian group": An abelian group that can be generated by a finite number of its elements. This is a key concept in classification theorems.
  • The structure theorem describes all finitely generated abelian groups.
• "Abelian group homomorphism": A function between two abelian groups that preserves the group operation.
  • The kernel of an abelian group homomorphism is itself an abelian group.

Variants and Related Words

• Abelian (adjective): Describes the property of being commutative. It can modify other nouns like "category" or "variety."
  • The category of abelian groups has nice properties.
• Commutative group (noun): A less common synonym for "abelian group."
• Group (noun): The more general algebraic structure of which an abelian group is a specific type.

Synonyms

• Commutative group: A direct synonym emphasizing the commutative property.

Notes on Meaning

• The concept is named after the Norwegian mathematician Niels Henrik Abel.
• The commutativity condition is expressed as: for all elements and in the group, + = + (if the operation is denoted as addition).

Noun

1. a group that satisfies the commutative law