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1.
A graph Γ is said to be symmetric if its automorphism group Aut(Γ)acts transitively on the arc set of Γ.We show that if Γ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group G of automorphisms,then either G is normal in Aut(Γ),or Aut(Γ)contains a non-abelian simple normal subgroup T such that G≤T and(G,T)is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups.If G is arc-transitive,then G is always normal in Aut(r),and if G is regular on the vertices of Γ,then the number of possible exceptional pairs(G,T)is reduced to 5. 相似文献
2.
M.L. Lewis 《Archiv der Mathematik》2000,75(5):338-341
We say that the degree graph G\Gamma has bounded Fitting height if there is a bound on the Fitting heights of the solvable groups for which G\Gamma is the degree graph. In this paper, we determine which degree graphs have bounded Fitting height. 相似文献
3.
Claudio Paolo Morresi Zuccari 《Journal of Algebra》2012,353(1):22-30
Let Γ be a graph in which each vertex is non-adjacent to another different one. We show that, if G is a finite solvable group with abelian Fitting subgroup and with character degree graph , then G is a direct product of subgroups having a disconnected character degree graph. In particular, Γ is a join of disconnected graphs. We deduce also that solvable groups with abelian Fitting subgroup have a character degree graph with diameter at most 2. 相似文献
4.
Alexander Moretó 《Algebras and Representation Theory》2007,10(4):333-338
Given a group G, Γ(G) is the graph whose vertices are the primes that divide the degree of some irreducible character and two vertices p and q are joined by an edge if pq divides the degree of some irreducible character of G. By a definition of Lewis, a graph Γ has bounded Fitting height if the Fitting height of any solvable group G with Γ(G)=Γ is bounded (in terms of Γ). In this note, we prove that there exists a universal constant C such that if Γ has bounded Fitting height and Γ(G)=Γ then h(G)≤C. This solves a problem raised by Lewis.
Research supported by the Spanish Ministerio de Educación y Ciencia, MTM2004-06067-C02-01 and MTM2004-04665, the FEDER and
Programa Ramón y Cajal. 相似文献
5.
Bounding the Fitting height of a solvable group by the number of zeros in a character table 总被引:1,自引:0,他引:1
Guohua Qian 《Proceedings of the American Mathematical Society》2002,130(11):3171-3176
In this paper, we bound the Fitting height of a solvable group by the number of zeros in a character table.
6.
We say that a group G satisfies the prime-power hypothesis if the GCDs for all pairs of distinct character degrees are prime powers. We prove that if G is a solvable group satisfying the prime-power hypothesis, then G has Fitting height at most 12. If in addition |G| is odd, then we prove that the Fitting height of G is at most 6. 相似文献
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Ahmet Arıkan 《代数通讯》2013,41(10):3643-3657
Call a group G hypersolvable if it has an ascending series with G/CG(A) solvable for each factor A of the series. In this article we establish some basic facts about hypersolvable groups. We also prove that if G is a perfect Fitting p-group such that every proper subgroup is contained in a proper normal subgroup, then G has a proper non-hypersolvable subgroup. 相似文献
9.
设G为有限群,cd(G)表示G的所有复不可约特征标次数的集合.本文研究了不可约特征标次数为等差数的有限可解群,得到两个结果:如果cd(G)={1,1+d,1+2d,…,1+kd},则k≤2或cd(G)={1,2,3,4};如果cd(G)={1,a,a+d,a+2d,…,a+kd},|cd(G)|≥4,(a,d)=1,则cd(G)={1,2,2e+1,2e+1,2(e+1)},并给出了d>1时群的结构. 相似文献
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11.
图G的Mostar指数定义为Mo(G)=∑uv∈Ε(G)|nu-nv|,其中nu表示在G中到顶点u的距离比到顶点v的距离近的顶点个数,nv表示到顶点v的距离比到顶点u的距离近的顶点个数.若一个图G的任两点之间的距离至多为2,且不是完全图,则称G是一个直径为2的图.已知直径为2点数至少为4的极大平面图的最小度为3或4.本文研究了直径为2且最小度为4的极大平面图的Mostar指数.具体说,若G是一个点数为n,直径为2,最小度为4的极大平面图,则(1)当n≤12时,Mostar指数被完全确定;(2)当n≥13时,4/3n2-44/3n+94/3≤Mo(G)≤2n2-16n+24,且达到上,下界的极图同时被找到. 相似文献
12.
A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given. 相似文献
13.
Let G be a finite group possessing a Carter subgroup K. Denote by \(\mathbf {h}(G)\) the Fitting height of G, by \(\mathbf {h}^*(G)\) the generalized Fitting height of G, and by \(\ell (K)\) the number of composition factors of K, that is, the number of prime divisors of the order of K with multiplicities. In 1969, E. C. Dade proved that if G is solvable, then \(\mathbf {h}(G)\) is bounded in terms of \(\ell (K)\). In this paper, we show that \(\mathbf {h}^*(G)\) is bounded in terms of \(\ell (K)\) as well. 相似文献
14.
Let G be a group. We consider the set cd(G)/{m}, where m ∈ cd(G). We define the graph △(G - m) whose vertex set is p(G - m), the set of primes dividing degrees in cd(G)/{m}. There is an edge between p and q in p(G - m) ifpq divides a degree a ∈ cd(G)/{m}. We show that if G is solvable, then △(G - m) has at most two connected components. 相似文献
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Yasuo Teranishi 《Linear and Multilinear Algebra》2013,61(6):577-585
Let G be the automorphism group of a graph Γ and let λ be an eigenvalue of the adjacency matrix of Γ. In this article, (i) we derive an upper bound for rank(G), (ii) if G is vertex transitive, we derive an upper bound for the extension degree of ?(λ) over ?, (iii) we study automorphism groups of graphs without multiple eigenvalues, (iv) we study spectra of quotient graphs associated with orbit partitions. 相似文献
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Let δ, γ, i and α be respectively the minimum degree, the domination number, the independent domination number and the independence number of a graph G. The graph G is 3-γ-critical if γ = 3 and the addition of any edge decreases γ by 1. It was conjectured that any connected 3-γ-critical graph satisfies i = γ, and is hamiltonian if δ ≥ 2. We show here that every connected 3-γ-critical graph G with γ ≥ 2 satisfies α ≤ δ + 2; if α = δ + 2 then i = γ; while if α ≤ δ + 1 then G is hamiltonian. © 1997 Wiley & Sons, Inc. J Graph Theory 25: 173–184, 1997 相似文献
19.
Péter Pál Pálfy 《Periodica Mathematica Hungarica》1998,36(1):61-65
For any finite solvable group G we show that if three primes dividing the degrees of certain irreducible characters of G are
given, then there exists an irreducible character of G with degree divisible by at least two of the given primes.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
20.
Let Σ be a (connected) surface of “complexity” κ; that is, Σ may be obtained from a sphere by adding either ½κ handles or κ crosscaps. Let ρ ≥ 0 be an integer, and let Γ be a “ρ-representative drawing” in Σ; that is, a drawing of a graph in Σ so that every simple closed curve in Σ that meets the drawing in < ρ points bounds a disc in Σ. Now let Γ′ be another drawing, in another surface Σ′ of complexity κ′, so that Γ and Γ′ are isomorphic as abstract graphs. We prove that. (i) If ρ ≥ 100 log κ/ log log κ (or ρ ≥ 100 if κ ≤ 2) then κ′ ≥ κ, and if κ′ = κ and Γ is simple and 3-connected there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, and. (ii) if Γ is simple and 3-connected and Γ′ is 3-representative, and ρ ≥ min (320, 5 log κ), then either there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, or κ′ ≥ κ + 10-4 ρ2. © 1996 John Wiley & Sons, Inc. 相似文献