关于类置换群和阶映射的两个猜想

发布时间：2018-04-09 01:20

  本文选题：置换矩阵群　切入点：类置换矩阵群　出处：《华中师范大学》2016年博士论文

【摘要】：本文研究类置换矩阵群的一个猜想和阶映射的一个猜想.我们首先考虑类置换矩阵群的猜想.如果一个矩阵群G中的每一个矩阵都相似于一个置换矩阵,那么称G为类置换矩阵群(permutation-like matrix group)如果存在一个可逆矩阵Q,使得对群G的每一个矩阵A, Q-1 AQ都是一个置换矩阵,那么称g是一个置换矩阵群(permutation matrix group)2005年,G.Cigler指出类置换矩阵群一般不是置换矩阵群.他提出一个猜想：猜想1.如果一个有限类置换矩阵群g包含一个极大循环,那么g是一个置换矩阵群.这里“极大循环”是指对应于长度等于维数的循环置换的置换矩阵G.Cigler证明了：如果n维类置换矩阵群G包含一个极大循环C使得C生成的子群正规于G,而且n=4或者n是素数,那么g是置换矩阵群.这个问题直到我们的结论(2014年)出来之前,没有任何进展.本文将证明：如果n维的类置换矩阵群G包含了一个极大循环C使得C生成的子群正规于G,而且n是一个素数的幂,那么G是置换矩阵群.第二部分内容是关于阶映射猜想.设有限群G与循环群C的阶都为n.如果双射f：G→C使得对任意9∈G,g的阶都是f(g)的阶的因数,那么我们就称f是从G到同阶循环群C的一个阶映射.如果存在这样的阶映射,就说G有阶映射.很多研究问题和结果与阶映射有密切联系,而且揭示下面的猜想有可能是正确的.猜想2.有限群G都有阶映射.本文将证明：如果G是有限可解群,那么G有阶映射.
[Abstract]:In this paper, we study a conjecture for a group of permutation matrices and a conjecture for a mapping of order.We first consider the conjecture of class permutation matrix groups.If every matrix in a matrix group G is similar to a permutation matrix, then G is called a permutation matrix group like matrix group. If there is an invertible matrix Q such that every matrix of group G A, Q-1 AQ is a permutation matrix.In 2005, G. Cigler pointed out that the group of permutation matrices is generally not a group of permutation matrices.He put forward a conjecture: conjecture 1.If a finite class permutation matrix group g contains a maximal cycle then g is a permutation matrix group.Here "maximal cycle" refers to a permutation matrix G.Cigler corresponding to a cyclic permutation of length equal to dimension. It is proved that if the n-dimensional permutation matrix group G contains a maximal cycle C such that the subgroup generated by C is normal to G, and nn 4 or n is a prime number,Then g is a group of permutation matrices.No progress has been made until our conclusion (2014) is reached.In this paper, we will prove that if n-dimensional class permutation matrix group G contains a maximal cycle C such that the subgroup generated by C is normal to G and n is a power of a prime number, then G is a permutation matrix group.The second part is about conjecture of order mapping.Let the order of finite group G and cyclic group C be n.If a bijection f: G G C such that the order of any 9 鈭，

本文编号：1724208