group theory
Noun: - The branch of mathematics dealing with groups: Group theory is a field of abstract algebra that studies algebraic structures known as groups. A group is a set equipped with a single binary operation that satisfies four fundamental properties: closure, associativity, the existence of an identity element, and the existence of inverse elements for all elements in the set. This theory explores the properties, structures, and applications of these mathematical objects.
- Noun:
- Understanding symmetry in crystals requires a solid foundation in group theory.
- The mathematician published a groundbreaking paper in the field of group theory.
- Many problems in particle physics are analyzed using the principles of group theory.
- "to apply group theory to": to use the concepts and methods of group theory in a different field.
- Researchers often apply group theory to problems in cryptography and coding theory.
- "a concept in group theory": a specific idea or theorem within the discipline.
- The notion of a cyclic subgroup is a fundamental concept in group theory.
- Group (n): In mathematics, the fundamental algebraic structure studied in group theory, consisting of a set and an operation.
- Group-theoretic (adj): Relating to or using the methods of group theory.
- The proof involved a sophisticated group-theoretic argument.
- Algebraic group theory: A more specific term often used when emphasizing the algebraic, rather than geometric, aspects of the field.
(Note: As a specialized academic noun, "group theory" does not have phrasal verbs. Common phrases involve its application.) - Study of groups: A descriptive phrase synonymous with group theory. - His doctoral research focused on the study of groups of prime order.
(Note: As a technical term, "group theory" is not used in idiomatic expressions. Its usage is confined to academic and scientific contexts.)
-
the branch of mathematics dealing with groups