  Mutually unbiased bases (MUB) are useful in a number of research areas. The
symmetry of MUB is an elusive and interesting subject. A (complete set of) MUB
in dimension $d$ is sharply covariant if it can be generated by a group of
order $d(d+1)$ from a basis state. Such MUB, if they exist, would be most
appealing to theoretical studies and practical applications. Unfortunately,
they seem to be quite rare. Here we prove that no MUB in odd prime dimensions
is sharply covariant, by virtue of clever applications of Mersenne primes,
Galois fields, and Frobenius groups. This conclusion provides valuable insight
about the symmetry of MUB and the geometry of quantum state space. It
complements and strengthens the earlier result of the author that only two
stabilizer MUB are sharply covariant. Our study leads to the conjecture that no
MUB other than those in dimensions 2 and 4 is sharply covariant.
