abelian group
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Definition
- 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
- a group that satisfies the commutative law