I need clarification on the definition of the order of the element in a group.
So this is the whole paragraph in my book leading to the definition:
"Let G be a group, and 'a' an element in G. Let us observe that "if there exists a nonzero integer m such that a^m=e, then there exists a positive integer n such that a^n=e."
If a^m=e where m is negative then a^-m=(a^m)^-1=e^-1=e. Thus, a^-m=e where -m is positive. This simple observation is crucial in our next definition. Let G be a arbitrary group , and 'a' a element in G:
Definition, If there exists a nonzero integer such that a^m=e, then the order of he element 'a' is defined to be the least positive integer 'n' such that a^n=e.
If there does not exist any nonzero integer m such that a^m=e, we say a has order infinity."
Ok, so i understand the concept, for example in Z_6, the order of 2, is 3 since '2+2+2=0' and e=0. But what I'm not understanding is what does variable 'm' have to do with anything?
Are we saying that for example 2+2+2=-2-2-2 ? But we cant cant say the order is -3 cause its not the least positive integer?