2.6. If G is a group in which (ab)i = aibi for three consecutive integers i for all a, b ∈ G, show that G is abelian.
Observe that if there exist two consecutive integers , such that and for all , then . Then we obtain . Now by multiplying this equation from left by and right by we obtain .
In our case taking and , we have and by taking and we have .
This shows that and now multiplying from right by we obtain . Hence is abelian
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