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

Abstract

If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. 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.

Analysis on the Properties of a Permutation Group

“…Take for instance, the elements of order 2 in the octahedron group O. It can be found that the elements in the class [2] and the class [22] are all the elements of order 2. However, the elements in the class [2] represent the rotation around the 2fold axes connecting the midpoints of two opposite edges, while the elements in the class [22] represent the rotations around the three coordinate axes through the angle π respectively.…”

“…It can be found that the elements in the class [2] and the class [22] are all the elements of order 2. However, the elements in the class [2] represent the rotation around the 2fold axes connecting the midpoints of two opposite edges, while the elements in the class [22] represent the rotations around the three coordinate axes through the angle π respectively. The permutation subgroup which is isomorphic to the group O is {E, (123), (132), (234), (243), (124), (142), (134), (143), (12) (34), (13) (24), (14) (23), (12), (13), (14) To summarize, on the basis of theoretical calculation and analysis, there are 11300 subgroups of the permutation group S 7 .…”

“…The study of permutation groups is of significance for the development of group-theoretical approach [1, 2,3]. On one hand, each row in the multiplication table of a finite group shows a permutation of group elements such that every finite group is a subgroup of a permutation group.…”

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

“…For any complex matrix A, by char A (x) we denote the characteristic polynomial of A, i.e., char A (x) = det(xI − A), where I denotes the identity matrix. [6,Lemma 2.3]). The following two are equivalent to each other:…”

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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.

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