Permutation-like matrix groups

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

“…Cigler [2,3] showed that a permutation-like matrix group is not a permutation matrix group in general. A d × d matrix is called a maximal cycle if it is similar to a permutation matrix corresponding to the cycle permutation of length d. Cigler conjectured that:…”

“…Cigler [2,3] proved that, if a permutation-like matrix group G of dimension prime p contains a maximal cycle which generates a normal cyclic subgroup, then G is a permutation matrix group. In [4], we further proved that this result still holds if we replace the dimension prime p by dimension p 2 .…”

“…References [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.…”