  If every element of a matrix group is similar to a permutation matrix, then
it is called a permutation-like matrix group. References [4] and [5] showed
that, if a permutation-like matrix group contains a maximal cycle of length
equal to a prime or a square of a prime and the maximal cycle generates a
normal subgroup, then it is similar to a permutation matrix group. In this
paper, we prove that if a permutation-like matrix group contains a maximal
cycle of length equal to any power of any odd prime and the maximal cycle
generates a normal subgroup, then it is similar to a permutation matrix group.
