有循环极大子群的素数幂阶群的作用是边传递的图(Ⅰ)

Γ是一个有限的、单的、无向的且无孤立点的图, G是Aut(Γ)的一个子群.如果G在Γ的边集合上传递,则称Γ是G-边传递图.我们完全分类了当G为一个有循环的极大子群的素数幂阶群时的G-边传递图.这扩展了Sander的结果.本文仅给出其中的一种情况,即当G同构于群时,所有的G-边传递图.结果为,是G-边传递的当且仅当Γ为下列图之一     

有循环极大子群的素数幂阶群的作用是边传递的图(II)

假定Γ是一个有限的、单的、无向的且无孤立点的图,G是Aut(Γ)的一个子群.如果G在Γ的边集合上传递,则称Γ是G-边传递图.我们完全分类了当G为一个有循环的极大子群的素数幂阶群时的G-边传递图.结果为:设图Γ含有一个阶为pn(p是素数,n≥2)的自同构群,且G有一个极大子群循环,则Γ是G-边传递的,当且仅当Γ同构于下列图之一1)pmK1,pn-1-m,0≤m≤n-1;2)pmK1,pn-m,0≤m≤n;3)pmKp,pn-m-1,0≤m≤n-2;4)pn-mCpm,pm≥3,m＜n;5)2n-2K1,1;6)pn-1-mCpm,pm≥3,m≤n-1;7)2pn-mCpm,pm≥3,m≤n-1;8)2pn-mK1,pm,0≤m≤n;9)pn-mK1,2pm,0≤m≤n;10)pn-mK2,pm,0＜m≤n;11)C(2pn-m,1,pm);12)pkC(2pm-k,1,pn-m),0＜k＜m,0＜m≤n;13)(t-s,2m)C(2m 1/(t-s,2m),1,2n-1-m),其中0≤m≤n-1,2n-2(s-1)≡0(mod 2m),t≡1(mod 2),s(≠)t(mod 2m),1≤s≤2m,1≤t≤2n-1;14)∪p i=1 Ci p n-1,其中Ci p n-1=Ca1a1 [1 (i-1)pn-2]a 1 2[1 (i--1)p n-2]…a 1 (pn-1-1)[1 (i-1)p n-2]≌Cp n-1,i=1,2,…,p;15)∪2 i=1 Ci 2n-1,其中Ci 2n-1=Ca1a 1 [1 (i-1)(2n-2-1)]a1 2[1 (i-1)(2n-2-1)]…a1 (2n-1-1)[1 (i-1)(2n-2-1)]≌C2n-1,i=1,2.     

素数阶群理论的量词消去算法及其上界

在以前的一些工作中,作者已经证明语言(?)={+,0,e)上素数阶群的理论T有量词消去性质并研究了它的判定问题的复杂性.本文在此基础上将利用T的判定问题的复杂性结果给出理论T的量词消去的一个算法,同时给出该算法的复杂性上界.     

关于一类内交换群边传递的图

本文所指的图是有限的、单的、无向的且无孤立点,p是素数.G=〈a,b|a~(p~α)=b~(p~β)=c~p=1,[b,a]=c,[a,c]=[b,c]=1〉(α≥β,(α,β,p)≠(1,1,2))是一类内交换p-群.进一步获得了G的性质和关于G-边传递的图的完全分类.     

素数阶群理论的量词消去及复杂性

本文中我们将研究语言,上素数阶群理论Ｔ的量词消去及相应的复杂性．我们证明理论Ｔ有量词消去性质,并利用该性质给出理论Ｔ判定问题的一个复杂性上界．     

有限素数度弧正则图

如果一个图的全自同构群在其弧集上正则,则称此图为弧正则图.本文刻画素数度的立方自由阶弧正则图,证明任何素数度2倍奇立方自由阶弧正则图都是正规或二部正规Cayley图,且不存在任意素数度4倍奇立方自由阶的弧正则图,推广了一些已知的结果,得到阶为8倍奇平方自由阶素数度弧正则图的分类,并发现新的弧正则图类.此外,基于所得的结果,我们提出一个猜想和有待后续研究的一些问题.     

有限群中素数方幂阶子群的介数

本文利用基础代数中有关稳定子，陪集等理论，给出了有限群G中p^k阶子群个数的一个结果。     

有限局部环上酉群阶的计算

设K=F_(q^2),其特征为p, q=p^α,K有对合自同构ω:a→a^q. G是一个p 群，其阶为p^β, 群代数R=KG为一局部环. K的2阶自同构ω可延拓为R的一个2阶自同构，记为ω'，为方便，对任意a∈R, 记ω‘(a)为~a. R上2n级酉群定义为U_(2n)R={A∈GL_(2n)R|A(0,I^n,I^n,0)~A^t=(0,I^n,I^n,0)} 该文计算了U_(2n)R的阶.      

具有很多素数阶元的有限群

考察元素的阶如何影响有限群的结构是群论中的一个重要课题．本文研究存在一个正规子群N，N外的元素都是素数阶元的有阶群．主要利用熟知的Thompson的一个定理，获得了这样的有限群．     

4p~2阶4度半传递图

如果图X的全自同构群Aut(X)作用在其顶点集V(X)和边集E(X)上都是传递的,但作用在弧集Arc(X)上非传递,则称X是半传递图.研究了4p~2(p3且p≡-1(mod4))阶4度半传递图,确定了4p~2阶4度半传递图的连通性及其自同构群的阶.     

Graphs on which a group of order <Emphasis Type="Italic">pq</Emphasis> acts edge-transitively

Let F be a finite simple undirected graph with no isolated vertices. Let p, q be prime numbers with p≥q. We complete the classification of the graphs on which a group of order pq acts edge-transitively. The results are the following. If Aut（Г） contains a subgroup G of order pq that acts edge-transitively on F, then F is one of the following graphs： （1） pK1,1; （2） pqK1,1; （3） pgq,1; （4） qKp,1 （p 〉 q）; （5） pCq （q 〉 2）; （6） qCp （p 〉 q）; （7） Cp （p 〉 q = 2）; （8） Cpq; （9） （Zp, C） whereC={±r^μ ｜μ∈Zq} withq〉2, q｜（p-1） and r≠1≡r^q （modp）; （10） Kp,1 （p 〉 q）; （11） a double Cayley graph B（G,C） with C = {1-r^μ ｜ μ ∈ Zq} and r≠1≡r^q （modp）; （12） Kpq,1;or （13） Kp,q.     

Graphs on which a group of order pq acts edge-transitively

Let Γ be a finite simple undirected graph with no isolated vertices. Let p, q be prime numbers with p ≥ q. We complete the classification of the graphs on which a group of order pq acts edge-transitively. The results are the following. If Aut(Γ) contains a subgroup G of order pq that acts edge-transitively on Γ, then Γ is one of the following graphs: (1) pK 1,1 ; (2) pqK 1,1 ; (3) pK q,1 ; (4) qK p,1 (p q); (5) pC q (q 2); (6) qC p (p q); (7) C p (p q = 2); (8) C pq ; (9) (Z p , C) where C = {±rμ | μ∈ Z q } with q 2, q|(p-1) and r ≡ 1 ≡ r q (mod p); (10) K p,1 (p q); (11) a double Cayley graph B(G, C) with C = {1-r μ | μ∈ Z q } and r ≡ 1 ≡ r q (mod p); (12) K pq,1 ; or (13) K p,q .     

Characterizability of the Group ^2Dp（3） by its Order Components,Where p ≥ 5 is a Prime Number Not of the Form 2^m ＋ 1

The author will prove that the group ^2Dp（3） can be uniquely determined by its order components, where p ≠ 2^m ＋ 1 is a prime number, p ≥ 5. More precisely, if OC（G） denotes the set of order components of G, we will prove OC（G） = OC（^2Dp（3）） if and only if G is isomorphic to ^2Dp（3）. A main consequence of our result is the validity of Thompson＇s conjecture for the groups under consideration.     

Recognition of some symmetric groups by the set of the order of their elements

For a finite group G, let π_e(G) be the set of order of elements in G and denote S _n the symmetric group on n letters. We will show that if π_e(G ) = π_e(H), where H is S _p or S _p+1 and p is a prime with 50 < p < 100, then G ≅ H. This revised version was published online in August 2006 with corrections to the Cover Date.     

Tetravalent edge-transitive graphs of order p 2 q

A graph is called edge-transitive if its full automorphism group acts transitively on its edge set.In this paper,by using classification of finite simple groups,we classify tetravalent edge-transitive graphs of order p2q with p,q distinct odd primes.The result generalizes certain previous results.In particular,it shows that such graphs are normal Cayley graphs with only a few exceptions of small orders.     

On a certain sum-product estimate in fields of prime order

Let Fp be the finite field of p elements with p prime.If A is a subset of Fp and g is an element of F*p with order ν,then max{|A + g·A|,|A·A|} (ν/(ν + |A|2) )1/12|A|13/12.     

On the prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis>) where <Emphasis Type="Italic">p</Emphasis> > 3 is a prime number

Let G be a finite group. We define the prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. Recently M. Hagie [5] determined finite groups G satisfying Γ(G) = Γ(S), where S is a sporadic simple group. Let p > 3 be a prime number. In this paper we determine finite groups G such that Γ(G) = Γ(PSL(2, p)). As a consequence of our results we prove that if p > 11 is a prime number and p ≢ 1 (mod 12), then PSL(2, p) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph. The third author was supported in part by a grant from IPM (No. 84200024).     

Group rings which are G-Dedekind prime rings

Let R be a commutative Noetherian domain and A be a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and A when the group ring R[A] is a G-Dedekind prime ring.