![]() ![]() The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. One of the most important mathematical achievements of the 20th century was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 19, that culminated in a complete classification of finite simple groups. The early history of group theory dates from the 19th century. Group theory is also central to public key cryptography. ![]() Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. ![]() Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. In abstract algebra, group theory studies the algebraic structures known as groups. ![]()
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