Reaction between 1-methylcyclopentanol and potassium permanganate
There is no reaction. 
t-BuOH may be used as a solvent for permanganate reactions. For example, see Abiko et al. [1].

References


Abiko, A.; Roberts, J. C.; Takemasa, T.; Masamune, S. $\ce{KMnO4}$ Revisited: Oxidation of Aldehydes to Carboxylic Acids in the Tert-Butyl Alcohol - Aqueous $\ce{NaH2PO4}$ System. Tetrahedron Letters 1986, 27 (38), 4537–4540. https://doi.org/10/c3t9xj.