Permutation-like matrix groups with a maximal cycle of power of odd prime length

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. Refs. [3] and [4] 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.