Answer to Question #143114 in Abstract Algebra for nan

Question #143114
if a is an element of a group of finite order, prove that a^m ≠ a^n whenever m ≠ n
1
Expert's answer
2020-11-09T20:07:09-0500

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."



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS