Prerogative is an inherited privilege or official right that one has over others. It is particularly related to monarchs and rulers' right to command, direct, rule and control.
"One could have thought it was Adam's prerogative to change Eve's mind."
Perogative is the misspelling of the above word and even most times remains its mispronunciation. It is not recognized in any dictionary as a variant of 'prerogative'.