somebody tell the ans: show that a^n * b^n = (aXb)^n for all a,b are elements of Group G when (G,*) is abelian

Answer 1

Induction: It is clearly true for n=1. Now assume it is true for some n. Then
a^(n+1) b^(n+1)=a*a^n*b^n *b
=a*(a*b)^n *b (Inductive assumption)
=(a*b)^n *a*b (Abelian)
=((a*b)^(n+1).

Answer 2

because G is abelian you can reorder any product of a,b to : a*b * a*b * a*b

Answer 3

i need a paper to show that:

a*a*a*a*a*....(n times) *b*b*b*b*...(n times)

For each time pick one a and one b,it will be equal to=

ab*ab*ab*ab....(n times) = (a*b)^n