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

TL;DR

This paper proves that permutation-like matrix groups containing a maximal cycle of length a power of two and generating a normal subgroup are similar to permutation matrix groups, extending previous results for odd prime powers.

Contribution

It establishes the similarity of permutation-like matrix groups with maximal cycles of length a power of two to permutation matrix groups, filling a gap in existing classifications.

Findings

Permutation-like matrix groups with maximal cycle length a power of two are similar to permutation matrix groups.

Extends previous results from odd prime powers to powers of two.

Provides a complete characterization for these matrix groups.

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], [5] and [6] showed that, if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals to a prime, or a square of a prime, or a power of an odd prime, then the permutation-like matrix group is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals to any power of 2, then it is similar to a permutation matrix group.