Let G be a group of finite order and "a" be arbitrary element of G. Then G contains finite number of elements. Consider the following powers of "a:"
"a^0=e, a, a^2,...a^k,..."
Since G is a group, it contains all powers of "a". Taking into account that exponenets of the degrees of the element "a" are infinitely many and group G contains only finite number of elements, we conclude that "a^n=a^m" for some "n\\ne m."
Consequently, it is not true that "a^m \u2260 a^n" whenever "m \u2260 n."
Comments
Leave a comment