Final answer:
To show that (xy)^n = x^n y^n for any integer n in an Abelian group, we use induction on n for positive integers and the definition of the inverse for negative integers, utilizing the commutative property of the group.
Step-by-step explanation:
To show that (xy)^n = x^n y^n for any integer n in an Abelian group G, where x and y are elements of G, we will use the property of commutativity since in an Abelian group, the order of multiplication does not affect the result.
First, we know that in an Abelian group, for any two elements a, b in G, it is true that ab = ba.
Next, we will proceed by induction on n:
- Base Case: For n=1, (xy)^1 = xy = x^1y^1, which is true by definition.
- Inductive Step: Assume the statement is true for some positive integer k, such that (xy)^k = x^k y^k. Now we need to prove it for k+1.
Consider (xy)^(k+1) = (xy)^k(xy) = x^k y^k xy by inductive hypothesis.
Since the group is Abelian, we can rearrange the terms to get x^k x y^k y = x^(k+1) y^(k+1).
Now for any negative integer n, we can take n = -m where m is positive. We have (xy)^(-m) = ((xy)^m)^-1 and by our earlier argument (xy)^m = x^m y^m. Therefore, (xy)^(-m) = (x^m y^m)^-1 = (y^m)^-1 (x^m)^-1 = y^-m x^-m = x^-m y^-m considering the group is Abelian.
This demonstrates that (xy)^n = x^n y^n holds for any integer n.