Invertible Matrices over Finite Additively Idempotent Semirings

We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of invertible matrices.     

Groups of Permutation Polynomials over Finite Fields

LetFbe a finite field. We apply a result of Thierry Berger (1996,Designs Codes Cryptography,7, 215–221) to determine the structure of all groups of permutations onFgenerated by the permutations induced by the linear polynomials and any power map which induces a permutation onF. This generalizes a result of Leonard Carlitz (1953,Proc. Amer. Math. Soc.,4, 538).     

有限环上典型群的阶

本文通过“取模”，“取值和”，“取积”等方法，将有单位元的有限环Ｒ上典型群阶的计算转化为有限域上典型群阶的计算，并计算了Ｒ上ｎ－维自由模Ｖ_ｎ（Ｒ）中ｋ－维自由子模的个数．     

q-Rook Polynomials and Matrices over Finite Fields

Connections betweenq-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel'sq-hit polynomial. Both this new statisticmatand another statistic for theq-hit polynomial ξ recently introduced by Dworkin are shown to induce different multiset Mahonian permutation statistics for any Ferrers board. In addition, for the triangular boards they are shown to generate different families of Euler–Mahonian statistics. For these boards the ξ family includes Denert's statisticden, and gives a new proof of Foata and Zeilberger's Theorem that (exc, den) is equidistributed with (des, maj). Thematfamily appears to be new. A proof is also given that theq-hit polynomials are symmetric and unimodal.     

Identities of Semigroups of Triangular Matrices over Finite Fields

It is proved that the semigroup of all triangular n × n matrices over a finite field K is inherently nonfinitely based if and only if n > 3 and |K|> 2.     

Flat Modules over Group Rings of Finite Groups

Let k be a commutative ring of coefficients and G be a finite group. Does there exist a flat k G-module which is projective as a k-module but not as a k G-module? We relate this question to the question of existence of a k-module which is flat and periodic but not projective. For either question to have a positive answer, it is at least necessary to have |k| ≥ ?_ω. There can be no such example if k is Noetherian of finite Krull dimension, or if k is perfect.     

Hereditarily Finitely Based Semigroups of Triangular Matrices over Finite Fields

Denote by the semigroup of all n×n upper triangular matrices over the field F _q of order q. It is shown that the variety generated by is hereditarily finitely based if and only if n≤2. Moreover, the semigroup is not hereditarily finitely based for any n≥3 and any q≥2.     

有限域上正交循环矩阵的构造和计数

<正>交循环矩阵在组合、编码等诸多研究领域具有重要的应用价值.运用提升理想和傅里叶反演变换等方法,给出了任意有限域上任意阶正交循环矩阵的构造方法,以及精确的计数结果.区别于已有的构造方法,此提升方法不需要分解模多项式.     

有限交换环上典型群的Carter子群

令R为有限交换局部环,K为其剩余类域,令|K|=q.本文研究了R上辛群Sp2nR和正交群O2nR的Carter子群的存在性及结构,并给出R上正交群O2nR在q≡-1（mod 4）情况下的Sylow 2-子群的正确描述.     

有限交换环上典型群的Sylow子群

令Ｒ为有限交换局部环，Ｍ表其唯一的极大理想，ｋ表其剩余类域．本文定出了Ｒ上的一般线性群ＧＬｎＲ，辛群Ｓｐ２ｎＲ及双曲正交群Ｏ２ｎＲ的Ｓｙｌｏｗ子群．一般讲，若ｃｈａｒｘ＝ｐ，上述三类典型群的Ｓｙｌｏｗ ｐ－子群分别同构于由某些特殊形式的矩阵生成的子群；若ｃｈａｒｋ≠ｐ，上述三类典型群的Ｓｙｌｏｗｐ－子群分别同构于一循环群或半二面体群与若干Ｚｐ型循环群的圈积。     

有限交换环上线性群的Carter子群

令Ｒ为有限交换局部环，Ｋ为其剩余类域．本文研究了Ｒ上一般线性群ＧＬｎＲ的Ｃａｒｔｅｒ子群的存在性及结构．得到的结果是：若ｃｈａｒＫ为奇数或Ｋ＝Ｆ２，ＧＬｎＲ中存在唯一的Ｃａｒｔｅｒ子群的共轭类，即Ｓｙｌｏｗ－２子群的正规化子；若ｃｈａｒＫ＝２且｜Ｋ｜＞２，ＧＬｎＲ中不含Ｃａｒｔｅｒ子群．     

On the Representation of Finite Rings by Matrices over Commutative Rings

In this paper, it is shown that all finite associative rings satisfying the identities nx?=?0 and x ³ f(x)?+?x ²?=?0, where n is an odd natural number and f(x) ∈ ?[x], are embeddable in the ring of matrices over some suitable commutative ring.     

Locally Finite Groups All of Whose Subgroups are Boundedly Finite over Their Cores

For n a positive integer, a group G is called core-n if H/H_Ghas order at most n for every subgroup H of G (where H_G is thenormal core of H, the largest normal subgroup of G containedin H). It is proved that a locally finite core-n group G hasan abelian subgroup whose index in G is bounded in terms ofn. 1991 Mathematics Subject Classification 20D15, 20D60, 20F30.     

Orthogonality Matrices for Modules over Finite Frobenius Rings and MacWilliams' Equivalence Theorem

MacWilliams' equivalence theorem states that Hamming isometries between linear codes extend to monomial transformations of the ambient space. One of the most elegant proofs for this result is due to K. P. Bogart et al. (1978, Inform. and Control37, 19–22) where the invertibility of orthogonality matrices of finite vector spaces is the key step. The present paper revisits this technique in order to make it work in the context of linear codes over finite Frobenius rings.     

有限局部环上对称矩阵的计数定理(I)

设$A_{n}(R)$是有限局部环$Z/p^{k}Z$上$n$阶对称矩阵的集合, 这里$n\geq 2$. $p$是大于$2$素数, $p\equiv1({\rm mod}4)$ 且$k>1$. 通过确定有限局部环$Z/p^{k}Z$上对称矩阵的标准型, 计算出$A_{n}(R)$在线性群${\rm GL}_{n}(R)$作用下的轨道数, 从而计算出由特定对称矩阵确定的正交群的阶以及与特定对称矩阵在同一轨道的对称矩阵的阶.     

利用辛对合矩阵构造Cartesian认证码

利用辛对合矩阵构作了一个C artesian认证码,并且计算了其参数.假设编码规则是按均匀概率分布选取时,计算了模仿攻击成功的概率PI和替换攻击成功的概率PS.     

Hadamard Matrices and Dihedral Groups

Let D _2p be a dihedral group of order 2p, where p is an odd integer. Let ZD _2p be the group ring of D _2p over the ring Z of integers. We identify elements of ZD _2p and their matrices of the regular representation of ZD _2p . Recently we characterized the Hadamard matrices of order 28 ([6] and [7]). There are exactly 487 Hadamard matrices of order 28, up to equivalence. In these matrices there exist matrices with some interesting properties. That is, these are constructed by elements of ZD ₆. We discuss relation of ZD _2p and Hadamard matrices of order n=8p+4, and give some examples of Hadamard matrices constructed by dihedral groups.