Property relationg to non-ableian group $G$

Question

1. If $G$ is a group in which $(a\cdot b)^i=a^i\cdot b^i$ for three consecutive integers $i$ for all $a,b\in G$, show that $G$ is abelian.

2. Show that the conclusion of above problem does not follow if we assume the relation $(a\cdot b)^i=a^i\cdot b^i$ for just two consecutive integers.

I have solved the first spending about 3-4 hours. However, I cannot come up with the example to problem 2.

Can anyone please help with that?

Answer 1

There are non-commutative groups of exponent $3$, for instance the Heisenberg group of order $27$. In such a group $a^3=e$ and $a^4=a$ for all $a$.