Question

2.8. Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

Answer 1

We have $a^5=e$ and $a^3b=ba^3$. Applying $a^3$ to the second equation we obtain $a^3(a^3b)=a^3(ba^3)=(a^3b)a^3=(ba^3)a^3$ and hence $a^6b=ba^6$.

As $a^6=a^5\cdot a=ea=a$ we obtain $ab=ba$.