發(fā)布時間：2018-04-09 01:20

  本文選題：置換矩陣群　切入點：類置換矩陣群　出處：《華中師范大學(xué)》2016年博士論文

【摘要】：本文研究類置換矩陣群的一個猜想和階映射的一個猜想.我們首先考慮類置換矩陣群的猜想.如果一個矩陣群G中的每一個矩陣都相似于一個置換矩陣,那么稱G為類置換矩陣群(permutation-like matrix group)如果存在一個可逆矩陣Q,使得對群G的每一個矩陣A, Q-1 AQ都是一個置換矩陣,那么稱g是一個置換矩陣群(permutation matrix group)2005年,G.Cigler指出類置換矩陣群一般不是置換矩陣群.他提出一個猜想：猜想1.如果一個有限類置換矩陣群g包含一個極大循環(huán),那么g是一個置換矩陣群.這里“極大循環(huán)”是指對應(yīng)于長度等于維數(shù)的循環(huán)置換的置換矩陣G.Cigler證明了：如果n維類置換矩陣群G包含一個極大循環(huán)C使得C生成的子群正規(guī)于G,而且n=4或者n是素數(shù),那么g是置換矩陣群.這個問題直到我們的結(jié)論(2014年)出來之前,沒有任何進(jìn)展.本文將證明：如果n維的類置換矩陣群G包含了一個極大循環(huán)C使得C生成的子群正規(guī)于G,而且n是一個素數(shù)的冪,那么G是置換矩陣群.第二部分內(nèi)容是關(guān)于階映射猜想.設(shè)有限群G與循環(huán)群C的階都為n.如果雙射f：G→C使得對任意9∈G,g的階都是f(g)的階的因數(shù),那么我們就稱f是從G到同階循環(huán)群C的一個階映射.如果存在這樣的階映射,就說G有階映射.很多研究問題和結(jié)果與階映射有密切聯(lián)系,而且揭示下面的猜想有可能是正確的.猜想2.有限群G都有階映射.本文將證明：如果G是有限可解群,那么G有階映射.
[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，

本文編號：1724208