You formulated the problem well, but very few lines of your solution attempt make sense (what is ??, why two symbols for an identity element?, what does the symbol "/" mean? ). In the language of groups, there are originally no 'powers'. is just our mere shorthand for with occurrences of , , with agreement that .
You included the definition of order of an element, so let's use it to solve our problem.
Let be the order of .
because group operation is associative. This shows that but , so we verified that which is the first requirement for to be the order of .
Now let's verify the second requirement. We argue by contradiction. Assume that there is a number , , such that . Then, in the same way as in the first step, we show that . But , thus . And this contradicts our assumption that is the order of .