Two Conjectures On Permutation-like Matrix Groups And Order Mapping

This paper is a research on two conjectures:a conjecture on permutation-like matrix group and a conjecture on order mapping.We first consider the conjecture on permutation-like matrix group. If every element in a matrix group G is similar to a permutation matrix, then we call G a permutation-like matrix group. If there exists an invertible matrix Q such that, for every A ∈G Q, Q-1AQ is a permutation matrix, then we say that G is equivalent to a permutation matrix group (G is a permutation matrix, for short). In 2005, G.Cigler pointed out, in general, a permutation-like matrix group is not equivalent to a permutation matrix group. So G.Cigler made a conjecture:Conjecture 1:Let G be a finite permutation-like matrix group containing a maximal cycle matrix. Then G is a permutation matrix group. By "a maximal cycle matrix" we call a permutation matrix corresponding to a cycle permutation of length equal to the dimension of the matrix. Let G be a permutation-like matrix group containing a normal cyclic subgroup generated by a maximal cycle matrix. G.Cigler proved that G is a permutation matrix group if n=4 or n is a prime number.There is no progress on the conjecture until our resultwas represented (2014). Let G(?) GLn(C) be a permutation-like matrix group containing a normal maximal cycle subgroup generated by a maximal cycle matrix. The paper will show that G is a permutation matrix group if n is a power of prime.The second part of this paper is on the conjecture on order mapping. Let G be a finite group and C be a cycle group of order the same as the order of G. If there exists a bijection f from G onto C such that the order of g is a divisor of the order of f(g) for every g ∈G, we call f an order mapping from G to C. We say that G has an order mapping if the order mapping f exists. There are a lot of mathematical questions and results closely related to the order mapping, which suggested that the following conjecture is true.Conjecture 2:Any finite group has a order mapping.The paper will show that G has a order mapping if G is a finite solvable group.