Group (mathematics)
Definition and Explanation
In Mathematics, a group is a set equipped with a binary operation that combines any two of its elements to form a third element in such a way that four conditions called group axioms are satisfied. These conditions are closure, associativity, identity and invertibility.
Group Axioms
The four group axioms are the fundamental properties that a mathematical group must satisfy.
Closure
Closure refers to the property that the result of the operation, or the "product", of any two elements in the set is always an element of the set. For example, if the operation is addition, then if a and b are two elements in the set, the sum of a and b, denoted by a+b, is also an element of the set.
Associativity
The property of associativity means that when three elements of the set are combined, the way in which the elements are grouped does not matter. In other words, if a, b, and c are elements in the set, then (a*b)*c equals a*(b*c) for all a, b, and c in the set.
Identity Element
The identity element, or neutral element, is a special type of element in the set such that when it is combined with any element of the set, it leaves the other element unchanged. For example, if the operation is multiplication, the identity element is 1, because for any number a, 1*a = a*1 = a.
Invertibility
Invertibility refers to the property that for each element in the set, there is another element in the set such that when the two are combined, the result is the identity element. The other element is called the "inverse" of the original element.
Types of Groups
There are various types of groups in mathematics, each with its own unique properties and uses. Some of the most common types include abelian groups, cyclic groups, and symmetric groups.
Abelian Groups
An Abelian group, also known as a commutative group, is a group in which the binary operation is commutative. This means that for any two elements a and b in the group, a*b = b*a.
Cyclic Groups
A Cyclic group is a group that is generated by a single element. This means that every element in the group can be written as a power of one fixed element.
Symmetric Groups
The Symmetric group on a set is the group consisting of all permutations of the set. The operation in this group is the composition of permutations.
Applications of Group Theory
Group theory, the study of groups, is a central part of modern mathematics. Its applications can be found in many areas of mathematics and science, including algebra, number theory, and physics.
Algebra
In Algebra, groups are used to study symmetries and geometric structures. They play a crucial role in the classification of polynomial equations.
Number Theory
In Number Theory, groups are used to study the properties of integers and solve Diophantine equations.
Physics
In Physics, groups are used to describe the symmetry properties of physical systems, such as the rotational symmetry of a circle or the symmetry of a crystal lattice.
