PERIODIC GROUP Meaning and
Definition
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A periodic group refers to a mathematical concept within the field of group theory. In mathematics, a group is a set of elements with a binary operation that satisfies specific properties. A periodic group adds an additional criterion, namely that every element within the group has finite period or order.
The period or order of an element in a group is defined as the smallest positive integer n such that when the element is repeatedly operated on itself n times using the group's binary operation, the result is the identity element of the group. The identity element is an element that, when combined with any other element using the group operation, does not change the other element.
A periodic group is characterized by the fact that every element within the group has a finite period. This means that for any element in the group, there exists a positive integer n such that raising the element to the power of n gives the identity element.
Periodic groups have several important properties and characteristics that mathematicians study and explore. They provide insight into the structure and behavior of groups, and often serve as foundations for more complex mathematical theories. The study of periodic groups has applications in various areas of mathematics, such as algebra, number theory, and cryptography.
