If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.

Question

I need to prove the equivalence of the following two statements, if $G$ is a group:

$G$ is abelian $\Leftrightarrow$ for each $a,b$ in $G$: $(ab)^n = a^n b^n$ for each integer $n$

I have done the $\Rightarrow$ implication, but I have no idea how to approach $\Leftarrow$, any hints are very welcome.

I see, start with aˆ(-n+1) & bˆ(-n+1) on each side and work towards the "centre". Thank you @SimonS — Sara, Oct 10 '15 at 14:07

score 1 · Answer 1 · answered Oct 10 '15 at 14:08

1

Consider the case $n = 2$. We have

$$abab = aabb \Rightarrow a^{-1}ababb^{-1} = a^{-1}aabbb^{-1} \Rightarrow ba = ab$$

answered Oct 10 '15 at 14:08

Simon S

1 Answers1