I have a couple questions involving showing a group with certain properties is abelian. 1. For the first, I'm supposed to show that if some group G has the property that (ab)i=aibi for some three consecutive integers i and all a,b in G, then G must be abelian. Using (aba-1)i=abia-1=aibia-i, I've been able to show that a2b=ba2 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=a3b3, show G is abelian. By defining an automorphism on G by sending a to a3, I've been able to show every element has a unique cube root. Using this, I've shown a2b=ba2, as above. But now I'm stuck. Thanks in advance for any help.