You need to write down the definition. For , the expression is defined to be ( times). We also define and to be the inverse of .
Try using these to prove your statements.
How do you prove the following exponent laws for groups?
let x be an element of group G. Then for all integers m and n:
(i) x^-n = (x^-1)^n
(ii) x^m . x^n = x^(m+n)
(iii) x^m . x^n = x^n . x^m
(iv) (x^m)^n = x^(mn)
They seem self-evident, but I'm not quite sure how to prove them, especially when m and n are negative integers.
Personally, I would show that for some , or in a `natural' way. You can then use this isomorphism to prove your result (you need to work out what I mean by `natural' (that is, find the isomorphism), because this is really the thing you use; you can't use the fact that the are isomorphic, you use the actual isomorphism).