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

Permutation-like Matrix Groups with a Maximal Cycle of Power of Odd Prime Length