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1.
Let G be a finite group and N be a normal subgroup of G. Suppose that the set of G-conjugacy class sizes of N is {1, m, n}, with m?<?n and m does not divide n. In this paper, we show that N is solvable, and we determine the structure of these subgroups. 相似文献
2.
Let G be a finite group. A normal subgroup N of G is a union of several G-conjugacy classes, and it is called n-decomposable in G if it is a union of n distinct G-conjugacy classes. In this paper, we first classify finite non-perfect groups satisfying the condition that the numbers of conjugacy classes contained in its non-trivial normal subgroups are two consecutive positive integers, and we later prove that there is no non-perfect group such that the numbers of conjugacy classes contained in its non-trivial normal subgroups are 2, 3, 4 and 5. 相似文献
3.
A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case. 相似文献
4.
D. Bundy 《Journal of Combinatorial Theory, Series A》2006,113(6):995-1007
A necessary and sufficient condition is proven for the connectivity of commuting graphs C(G,X), where G is Sym(n), the symmetric group of degree n, and X is any G-conjugacy class. 相似文献
5.
Let N be a normal subgroup of a group G. The positive integers m and n are the two longest sizes of the non-central G-conjugacy classes of N with m > n and (m,n) = 1. In this paper, the structure of N is determined when n divides |N/N ∩ Z(G)|. Some known results are generalized. 相似文献
6.
I. B. Gorshkov 《Proceedings of the Steklov Institute of Mathematics》2016,293(1):58-65
Let G be a finite group G, and let N(G) be the set of sizes of its conjugacy classes. It is shown that, if N(G) equals N(Alt n ) or N(Sym n ), where n > 1361, then G has a composition factor isomorphic to an alternating group Altm with m ≤ n and the interval (m, n] contains no primes. 相似文献
7.
Let G be a finite group with a non-central Sylow r-subgroup R, Z(G) the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z(G)=1 and the G-conjugacy class size of every element of N is either 1 or m. 相似文献
8.
LetH be a subgroup of a groupG. A normal subgroupN H ofH is said to be inheritably normal if there is a normal subgroup N G of G such that N H = N G ∩ H. It is proved in the paper that a subgroup $N_{G_i }$ of a factor G i of the n-periodic product Π i∈I n G i with nontrivial factors G i is an inheritably normal subgroup if and only if $N_{G_i }$ contains the subgroup G i n . It is also proved that for odd n ≥ 665 every nontrivial normal subgroup in a given n-periodic product G = Π i∈I n G i contains the subgroup G n . It follows that almost all n-periodic products G = G 1 * n G 2 are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents. 相似文献
9.
Let G be a finite group, and let N(G) be the set of conjugacy class sizes of G. By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N(G) = N(L), then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In this article, we investigate validity of Thompson’s conjecture under a weak condition for the alternating groups of degrees p+1 and p+2, where p is a prime number. This work implies that Thompson’s conjecture holds for the alternating groups of degree p + 1 and p + 2. 相似文献
10.
Let G be a finite group and G?? be the set of primary, biprimary and triprimary elements of G. We prove that if the conjugacy class sizes of G?? are {1,m,n,mn} with positive coprime integers m and n, then G is solvable. This extends a recent result of Kong (Manatsh. Math. 168(2) (2012) 267–271). 相似文献
11.
For a graph G, ??(G) denotes the minimum degree of G. In 1971, Bondy proved that, if G is a 2-connected graph of order n and d(x)?+?d(y)????n for each pair of non-adjacent vertices x,y in G, then G is pancyclic or G?=?K n/2,n/2. In 2001, Xu proved that, if G is a 2-connected graph of order n????6 and |N(x)????N(y)|?+???(G)????n for each pair of non-adjacent vertices x,y in G, then G is pancyclic or G?=?K n/2,n/2. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if G is a 2-connected graph of order n????6 and |N(x)????N(y)|?+?d(w)????n for any three vertices x,y,w of d(x,y)?=?2 and wx or $wy\not\in E(G)$ in G, then G is 4-vertex pancyclic or G belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results. 相似文献
12.
Mingchun XU 《Frontiers of Mathematics in China》2013,8(5):1227-1236
For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson’s conjecture is true for the alternating group A 22 with connected prime graph. 相似文献
13.
D. B. Rokhlin 《Mathematical Notes》2007,81(3-4):543-548
For set-valued random sequences (G n) n=0 N with relatively open convex values G n(ω), we prove a new test for the existence of a sequence (x n) n=0 N of selectors adapted to the filtration and admitting an equivalent martingale measure. The statement is formulated in terms of the supports of regular upper conditional distributions of G n. This is a strengthening of the main result proved in our previous paper [1], where the openness of the set G n(ω) was assumed and a possible weakening of this condition was discussed. 相似文献
14.
B. Fleury 《Probability Theory and Related Fields》2012,152(1-2):141-178
Let K N = [±G 1, . . . , ±G N ] be the absolute convex hull of N independent standard Gaussian random points in ${\mathbb R^n}$ with N ≥ n. We prove that, for any 1-Lipschitz function ${f:\mathbb R^n\rightarrow\mathbb R}$ , the polytope K N satisfies the following Poincaré inequality in mean value: $$\mathbb {E}_{\omega} \int\limits_{K_N(\omega)} \left( f(x) - \frac{1}{\textup{vol}_n\left(K_N(\omega)\right)} \int\limits_{K_n(\omega)}f(y)dy \right)^2 dx \leq \frac{C}{n} \mathbb E_{\omega} \int\limits_{K_N(\omega)}|x|^2dx$$ where C?>?0 is an absolute constant. This Poincaré inequality is the one suggested by a conjecture of Kannan, Lovász and Simonovits for general convex bodies. Moreover, we prove in mean value that the volume of the polytope K N is concentrated in a subexponential way within a thin Euclidean shell with the optimal dependence of the dimension n. An important tool of the proofs is a representation of the law of (G 1, . . . , G n ) conditioned by the event that “the convex hull of G 1, . . . , G n is a (n ? 1)-face of K N ”. As an application, we also get an estimate of the number of (n ? 1)-faces of the polytope K N , valid for every N ≥ n. 相似文献
15.
Qingjun Kong 《Monatshefte für Mathematik》2012,168(2):267-271
Let G be a finite group and let G* be the set of elements of primary, biprimary and triprimary orders of G. We show that suppose that the conjugacy class sizes of G* are exactly {1, p a , n, p a n} with (p, n)?=?1 and a??? 0, then G is solvable. 相似文献
16.
Xing Huang 《印度理论与应用数学杂志》2015,46(6):773-780
Let id(v) denote the implicit degree of a vertex v in a graph G. We define G to be implicit 1-heavy (implicit 2-heavy) if at least one (two) of the end vertices of each induced claw has (have) implicit degree at least n/2. In this paper, we prove that: (a) Let G be a 2-connected graph of order n ≥ 3. If G is implicit 2-heavy and |N(u) ∩ N(v)| ≥ 2 for every pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian. (b) Let G be a 3-connected graph of order n ≥ 3. If G is implicit 1-heavy and |N(u) ∩ N(v)| ≥ 2 for each pair of vertices u and v with d(u, v) = 2 and max{id(u), id(v)} < n/2, then G is hamiltonian. 相似文献
17.
Gronwall’s function G is defined for n>1 by $G(n)=\frac{\sigma(n)}{n \log\log n}$ where σ(n) is the sum of the divisors of n. We call an integer N>1 a GA1 number if N is composite and G(N)≥G(N/p) for all prime factors p of N. We say that N is a GA2 number if G(N)≥G(aN) for all multiples aN of N. In (Caveney et al. Integers 11:A33, 2011), we used Robin’s and Gronwall’s theorems on G to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. In the present paper, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers ≤5040, and prove that a GA2 number N>5040 exists if and only if RH is false, in which case N is even and >108576. 相似文献
18.
For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A i ) is necessarily isomorphic to A i , where i ∈ {2p, 2p + 1}. 相似文献
19.
Joseph A. Wolf 《Mathematische Annalen》2011,349(2):263-299
In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G n /K n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N n nilpotent, in other words pairs (G n , K n ) for which G n /K n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L 2(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G n /K n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L 2(G/K). This allows an interpretation of L 2(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free. 相似文献
20.
Dessislava H. Kochloukova 《Journal of Pure and Applied Algebra》2006,204(3):536-554
A group G is knot-like if it is finitely presented of deficiency 1 and has abelianization G/G′?Z. We prove the conjecture of E. Rapaport Strasser that if a knot-like group G has a finitely generated commutator subgroup G′ then G′ should be free in the special case when the commutator G′ is residually finite. It is a corollary of a much more general result : if G is a discrete group of geometric dimension n with a finite K(G,1)-complex Y of dimension n, Y has Euler characteristics 0, N is a normal residually finite subgroup of G, N is of homological type FPn-1 and G/N?Z then N is of homological type FPn and hence G/N has finite virtual cohomological dimension vcd(G/N)=cd(G)-cd(N). In particular either N has finite index in G or cd(N)?cd(G)-1.Furthermore we show a pro-p version of the above result with the weaker assumption that G/N is a pro-p group of finite rank. Consequently a pro-p version of Rapaport's conjecture holds. 相似文献