The original question I was asking was Why does a body not rotate if force is applied on the centre of mass?.
The obvious answer is that the torque (R x F)${\bf r}\times{\bf F}$ is zero as R${\bf f}$ is zero at the centre of mass, but that defeats the purpose of my question. Why is the centre of mass so special that we take torque about it and force applied on it does not provide rotation?
I am looking for the intuition behind this.