Relationship between the power of a convex lens and the refractive index of a medium

According to the Lensmaker's formula , $\frac{1}{f} = (μ-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ Apparently, focal length $f$ is inversely proportional to the refractive index of the medium $μ$ . Since, Optical Power $P = \frac{1}{f}$ . This should imply that optical power is directly proportional to the refractive index. However, today I got told by my teacher that these two are independent. Where have I gone wrong in my reasoning? Is the fact provided to me correct and if yes, how?