Evaluate all order of every element of higher 100, 105 and 107 order of group for multiplication composition - Mannan, Md. Abdul; Bishnu, Kanchon Kumar; Islam, Shoma; Chowdhury, Md. Shafiul Alam; Hossain, Md. Amzad; Islam, Md. Shafikul; Khan, Sahib Jada Eyakub; Khatun, Hashiara (Edelweiss Applied Science and Technology, 2025)
This paper aims at treating a study on the order of every element of higher 100, 105, and 107 orders of group for multiplication composition. But the composition in G is associative; the multiplication composition is very significant in the order of elements of a group. We develop the order of a group o(G), higher order of groups in different types of order and the order of elements o(a) of a group in real numbers. Let G be a group and let a^n ∈ G be of infinite order n, then find Highest Common Factor (i.e., (n, m) denotes H.C.F of n and m). The Highest Common Factor of two numbers is the “smallest non-zero common number” which is a multiple of both the numbers. So o(a^m) = n / H.C.F(n, m), where (n, m) denotes the H.C.F of n and m. If a ∈ G is of order n, then there exists an integer m for which a^m = e if m is a multiple of n, in general we use this. Then we develop orders of elements of a cyclic group and every element of higher order of a group. After that we find out the order of every element of a group for the higher orders of the group for being binary operation.