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

TL;DR

This paper proves that permutation-like matrix groups containing a maximal cycle of any odd prime power length, which generates a normal subgroup, are similar to permutation matrix groups, extending previous results for prime and prime square lengths.

Contribution

It generalizes prior work by showing that such groups with maximal cycles of any odd prime power length are similar to permutation matrix groups.

Findings

Groups with maximal cycles of odd prime power length are similar to permutation matrix groups.

Normal subgroup generated by the maximal cycle is key to the similarity.

Extends previous results from prime and prime square lengths to all odd prime powers.

Abstract

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.