Evaluate All The Order of Every Element in The Higher Even, Odd, and Prime Order of Group for Composition

Abstract

This paper aims to treat a study on the order of every element in the higher even, odd and prime order of group for composition. In fact, express order of a group and order of an element of a group in real numbers. Here we discuss the higher order of groups in different types of order, which will give us practical knowledge to see the applications of the composition. In order to find out the order of an element am ∈G in which an = e= identity element, then find the least common multiple (i.e.(LCM))= λ) of m and n. The least common multiple of two numbers is the "smallest non-zero common number," which is a multiple of both the numbers. So O(am)= λ /m. Also, if G is a finite group, n is a positive integer, and a∈G then the order of the products na. When G is a finite group, every element must have finite order, but the converse is false. There are infinite groups where each element has finite order. Finally, find out the order of every element of a group in different types of the higher even, odd and prime order of group for composition.