An Abelian group is a set equipped with a binary operation that is associative, has an identity element, includes an inverse for each element, and is commutative. This structure is fundamental in abstract algebra and underpins many areas of mathematics, including number theory and topology.

Abelian Group

Abelian group

A binary operation is a calculation that combines two elements (operands) to produce another element within the same set. It is fundamental in algebraic structures such as groups, rings, and fields, where it must satisfy specific properties like closure, associativity, identity, and invertibility depending on the structure.

binary operation

associative

An identity element in a mathematical structure is a special element that, when combined with any element of the structure under a given operation, leaves the other element unchanged. It is fundamental in defining algebraic structures like groups, rings, and fields, where it ensures the existence of a neutral element for the operation.

identity element

inverse for each element

commutative

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields, focusing on their properties and the relationships between them. It provides a unifying framework for understanding mathematical concepts and solving problems across various areas of mathematics and science.

abstract algebra

Number theory is a branch of pure mathematics devoted to the study of the integers and integer-valued functions, exploring properties such as divisibility, prime numbers, and the solutions to equations in integers. It has deep connections with other areas of mathematics and finds applications in cryptography, computer science, and mathematical puzzles.

number theory

Topology is a branch of mathematics that studies the properties of space that are preserved under continuous transformations such as stretching and bending, but not tearing or gluing. It provides a foundational framework for understanding concepts of convergence, continuity, and compactness in various mathematical contexts.

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