I have a couple questions involving showing a group with certain properties is abelian.(adsbygoogle = window.adsbygoogle || []).push({});

1. For the first, I'm supposed to show that if some group G has the property that (ab)^{i}=a^{i}b^{i}for some three consecutive integers i and all a,b in G, then G must be abelian. Using (aba^{-1})^{i}=ab^{i}a^{-1}=a^{i}b^{i}a^{-i}, I've been able to show that a^{2}b=ba^{2}for all a,b in G, but I can't get any farther.

2. The second is similar. Given that a finite group G has order not divisible by 3, and for every a,b in G, (ab)^{3}=a^{3}b^{3}, show G is abelian. By defining an automorphism on G by sending a to a^{3}, I've been able to show every element has a unique cube root. Using this, I've shown a^{2}b=ba^{2}, as above. But now I'm stuck.

Thanks in advance for any help.

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# Showing a group is abelian

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