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1.
OD-characterization of Almost Simple Groups Related to U3(5)   总被引:1,自引:0,他引:1  
Let G be a finite group with order |G|=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg- Kegel graph) denoted .by г(G) (or GK(G)). This graph is constructed as follows: The vertex set of it is π(G) = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge (and we write pi - pj) if and only if G contains an element of order pipj. The degree deg(pi) of a vertex pj ∈π(G) is the number of edges incident on pi. We define D(G) := (deg(p1), deg(p2),..., deg(pk)), which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that |H| = |G| and D(H) = D(G). Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L := U3(5) be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.  相似文献   

2.
The degree pattern of a finite group has been introduced in [18].A group M is called k-fold OD- characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M .In particular,a 1-fold OD-characterizable group is simply called OD-characterizable.It is shown that the alternating groups A m and A m+1 ,for m = 27,35,51,57,65,77,87,93 and 95,are OD-characterizable,while their automorphism groups are 3-fold OD-characterizable.It is also shown that the symmetric groups S m+2 ,for m = 7,13,19,23,31,37,43,47,53,61,67,73,79,83,89 and 97,are 3-fold OD-characterizable.From this,the following theorem is derived.Let m be a natural number such that m 100.Then one of the following holds: (a) if m = 10,then the alternating groups A m are OD-characterizable,while the symmetric groups S m are OD- characterizable or 3-fold OD-characterizable;(b) the alternating group A 10 is 2-fold OD-characterizable;(c) the symmetric group S 10 is 8-fold OD-characterizable.This theorem completes the study of OD-characterizability of the alternating and symmetric groups A m and S m of degree m 100.  相似文献   

3.
The degree pattern of a finite group G associated with its prime graph has been introduced by Moghaddamfar in 2005 and it is proved that the following simple groups are uniquely determined by their order and degree patterns:All sporadic simple groups,the alternating groups Ap(p≥5 is a twin prime)and some simple groups of the Lie type.In this paper,the authors continue this investigation.In particular,the authors show that the symmetric groups Sp+3,where p+2 is a composite number and p+4 is a prime and 97相似文献   

4.
A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G,in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F,the zeta function of F does not appear.In the present paper we describe effectively all exceptional groups of orders divisible by exactly two prime numbers p and q,which have unique subgroups of orders p and q.  相似文献   

5.
A RECOGNITION OF SIMPLE GROUPS PSL(3, q) BY THEIR ELEMENT ORDERS   总被引:2,自引:0,他引:2  
For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.  相似文献   

6.
Let G be a finite group.A subgroup H of G is called an H-subgroup in G if NG(H) ∩Hg≤H for all g∈G.A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G=HK and H∩K is an H-subgroup in G.In this paper,we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G.Our results improve and generalize several recent results in the literature.  相似文献   

7.
A k-subset D of a group G of order v (1 < k < v -1 ) is called a (v, k, )-difference set ifevery nonidentity element of G appears in the list of d1d2-1(d1, d2 D) exactly times. Setn = k - , n is called the order of difference set D.An taomorphism of a finite group G is called a (right) multiplier of a difference set D inG if sends D onto Da for some a G. If G is abelian and if is given by g gt, where t isan integer prime to the order of G, then and also t itself are called a numerical mul…  相似文献   

8.
Let G be a finite group. A Cayley graph over G is a simple graph whose automorphism group has a regular subgroup isomorphic to G. A Cayley graph is called a CI-graph(Cayley isomorphism) if its isomorphic images are induced by automorphisms of G. A well-known result of Babai states that a Cayley graph Γ of G is a CI-graph if and only if all regular subgroups of Aut(Γ) isomorphic to G are conjugate in Aut(Γ). A semi-Cayley graph(also called bi-Cayley graph by some authors) over G is a simple graph whose automorphism group has a semiregular subgroup isomorphic to G with two orbits(of equal size). In this paper, we introduce the concept of SCI-graph(semi-Cayley isomorphism)and prove a Babai type theorem for semi-Cayley graphs. We prove that every semi-Cayley graph of a finite group G is an SCI-graph if and only if G is cyclic of order 3. Also, we study the isomorphism problem of a special class of semi-Cayley graphs.  相似文献   

9.
Let σ = {σ_i | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G, for some i ∈ I, and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HA~x= A~xH for all A ∈ H and x ∈ G:σ-subnormal in G if there is a subgroup chain A = A_0≤A_1≤···≤ A_t = G such that either A_(i-1)■A_i or A_i/(A_(i-1))A_i is a finite σ_i-group for some σ_i ∈σ for all i = 1,..., t.If M_n M_(n-1) ··· M_1 M_0 = G, where Mi is a maximal subgroup of M_(i-1), i = 1, 2,..., n, then M_n is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal(σ-quasinormal,respectively) in G but, in the case n 1, some(n-1)-maximal subgroup is not σ-subnormal(not σ-quasinormal,respectively) in G, we write m_σ(G) = n(m_(σq)(G) = n, respectively).In this paper, we show that the parameters m_σ(G) and m_(σq)(G) make possible to bound the σ-nilpotent length l_σ(G)(see below the definitions of the terms employed), the rank r(G) and the number |π(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when m_σ(G) = |π(G)|. Some known results are generalized.  相似文献   

10.
Recognition of the Projective Special Linear Group over GF(3)   总被引:1,自引:0,他引:1  
M.  R.  DARAFSHEH 《数学学报(英文版)》2010,26(3):477-488
Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp(3), where p 〉 3 is a prime number, is at most 2-recognizable.  相似文献   

11.
对于任意一个有限群G,令π(G)表示由它的阶的所有素因子构成的集合.构建一种与之相关的简单图,称之为素图,记作Γ(G).该图的顶点集合是π(G),图中两顶点p,g相连(记作p~q)的充要条件是群G恰有pq阶元.设π(G)={P1,p2,…,px}.对于任意给定的p∈π(G),令deg(p):=|{q∈π(G)|在素图Γ(G)中,p~q}|,并称之为顶点p的度数.同时,定义D(G):=(deg(p1),deg(p2),…,deg(ps)),其中p12<…相似文献   

12.
Let G be a group and πe(G) the set of element orders of G.Let k∈πe(G) and m k be the number of elements of order k in G.Letτe(G)={mk|k∈πe(G)}.In this paper,we prove that L2(16) is recognizable byτe (L2(16)).In other words,we prove that if G is a group such that τe(G)=τe(L2(16))={1,255,272,544,1088,1920},then G is isomorphic to L2(16).  相似文献   

13.
设G是一个有限群,S是G的不包含单位元1的非空子集,定义群G关于S的Cayley(有向)图X:=Cay(G,S)如下:V(X)=G,E(X)={(g,sg)|g∈G,s∈S}.Cayley(有向)图X:=Cay(G,S)称为正规的,如果G的右正则表示R(G)在X的自同构群Aut(X)中是正规的.设G是4p阶二面体群(p为素数).考察了Cay(G,S)连通3度的正规性,并给出了这些图的全自同构群.  相似文献   

14.
Pm×Kn的邻点可区别全色数   总被引:6,自引:0,他引:6  
设G是简单图.设f是一个从V(G)∪E(G)到{1,2,…,k}的映射.对每个v∈V(G),令C_f(v)={f(v)}∪{f(vw)|w∈V(G),vw∈E(G)}.如果f是k-正常全染色,且对任意u,v∈V(G),uv∈E(G),有C_f(u)≠C_f(v),那么称f为图G的邻点可区别全染色(简称为k-AVDTC).数x_(at)(G)=min{k|G有k-AVDTC}称为图G的邻点可区别全色数.本文给出路P_m和完全图K_n的Cartesion积的邻点可区别全色数.  相似文献   

15.
图的无符号拉普拉斯矩阵是图的邻接矩阵和度对角矩阵的和,其特征值记为q1≥q2≥…≥qn.设C(n,m)是由n个顶点m条边的连通图构成的集合,这里1≤n-1≤m≤(n2).如果对于任意的G∈C(n,m)都有q1(G*)≥q1(G)成立,图G*∈C(n,m)叫做最大图.这篇文章证明了对任意给定的正整数a=m-n+1,如果n...  相似文献   

16.
The degree pattern of a finite group M has been introduced by A. R. Moghaddamfar et al. [Algebra Colloquium, 2005, 12(3): 431–442]. A group M is called k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and degree pattern as M. In particular, a 1-fold OD-characterizable group is simply called OD-characterizable. In this article, we will show that the alternating groups A p+3 for p = 23, 31, 37, 43 and 47 are OD-characterizable. Moreover, we show that the automorphism groups of these groups are 3-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.  相似文献   

17.
有限群的最大子群的性质对群结构的影响   总被引:1,自引:0,他引:1  
有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤HG=CoreG(H).本文证明了:设F是一个包含超可解群系u的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F*(H)的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.  相似文献   

18.
The prime graph of a finite group was introduced by Gruenberg and Kegel. The degree pattern of a finite group G associated to its prime graph was introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions (1) |G| = |H| and (2) D(G) = D(H). Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Till now a lot of finite simple groups were shown to be OD-characterizable, and also some finite groups especially the automorphism groups of some finite simple groups were shown not being OD-characterizable but k-fold OD-characterizable for some k > 1. In the present paper, the authors continue this topic and show that the automorphism groups of orthogonal groups O 10+(2) and O 10(2) are OD-characterizable.  相似文献   

19.
Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p 1, p 2,...,p k }, where p 1 < p 2 < ... < p k , then the degree pattern of G is defined by D(G) = (deg(p 1), deg(p 2),...,deg(p k )). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H| = |G| and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A 22 is OD-characterizable. We also show that the automorphism groups of the alternating groups A 16 and A 22, i.e., the symmetric groups S 16 and S 22 are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.  相似文献   

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