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In Abstract Algebra , a group is a Set with a Binary Operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called Group Theory . Many of the structures investigated in mathematics turn out to be groups. These include familiar number systems, such as the Integer s, the Rational Number s, the Real Number s, and the Complex Number s under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Other important examples are the group of non-singular Matrices under multiplication and the group of Invertible Functions under Composition . Group theory allows for the properties of such structures to be investigated in a general setting. Group theory has extensive applications in mathematics, science, and engineering. Many Algebraic Structure s such as Field s and Vector Space s may be defined concisely in terms of groups, and group theory provides an important tool for studying Symmetry , since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving Symmetry principles, such as Relativity , Quantum Mechanics , and Particle Physics . Furthermore, their ability to represent Geometric transformations finds applications in Chemistry , Computer Graphics , and other fields. HISTORY See Also: Group theory DEFINITIONS
Some texts omit the explicit requirement of closure, since the closure of the group follows from the definition of a Binary Operation . Using the identity element property it can be shown that a group has exactly one identity element. See Simple Theorems . The inverse of an element can also be shown to be unique, and the left- and right-inverses of an element are the same. Some definitions are thus slightly more narrow, substituting the second and third axioms with the concept of a "left (or right) identity element" and a "left (or right) inverse element."
BASIC CONCEPTS IN GROUP THEORY Order of groups and elements |
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