  We prove that a prime knot K is not determined by its p-fold cyclic branched
cover for at most two odd primes p. Moreover, we show that for a given odd
prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold
cyclic branched cover of at most one more knot K' non equivalent to K. To prove
the main theorem, a result concerning the symmetries of knots is also obtained.
This latter result can be interpreted as a characterisation of the trivial
knot.
