This paper aims at treating a study on the order of every element of 60 and 61 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, higher order of groups in different types of order and the order of elements of a group in real numbers. Let G be a group and let
be of infinite order n. In addition, notation na = e and n is a least positive integer
O(a) = n. If
is of order n, then there exists an integer m for which
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.
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