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1.
Let A m be the variety of abelian groups of exponent m, where mis a square free positive integer. It is shown that every group G has an A m -covering group, which generalizes the work of I. Schur (1904) and M.R. Jones (1973). Also, similar to the work of Yamazaki (1964) in the variety of abelian groups, if 1→ AHG → 1 is an A m -stem extension, then we show that H is a homomorphic image of an A m -covering group of G. Finally,in this variety some results of Hall's type will be proved.  相似文献   

2.
Let X be a nonempty subset of a group G. We call a subgroup A of G an Xmsemipermutable subgroup of G if A has a minimal supplement T in G such that for every maximal subgroup M of any Hall subgroup T1 of T there exists an element xX such that AMx = MxA. In this paper, we study the structure of finite groups with some given systems of Xm‐semipermutable subgroups (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

3.
Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups 2 G 2(q 2) for q 2 = 32m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many of the nonabelian simple groups.  相似文献   

4.
We prove that ifA≠1 is a subgroup of a finite groupG and the order of an element in the centralizer ofA inG is strictly larger (larger or equal) than the index [G:A], thenA contains a non-trivial characteristic (normal) subgroup ofG. Consequently, ifA is a stabilizer in a transitive permutation group of degreem>1, thenexp(Z(A))<m. These theorems generalize some recent results of Isaacs and the authors.  相似文献   

5.
Norm Inequalities for Commutators of Self-adjoint Operators   总被引:3,自引:0,他引:3  
Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with and for some real numbers a 1, a 2, b 1, and b 2, then for every unitarily invariant norm|||·|||,
. If, in addition, X is positive, then
. These norm inequalities generalize recent related inequalities due to Kittaneh, Bhatia-Kittaneh, and Wang-Du.   相似文献   

6.
M. González  J. Otal 《代数通讯》2013,41(10):3405-3412

Let A be an elementary abelian group of order at least p 3 acting on a finite p′-group G that is soluble with derived length d. Assume that γ c (C G (a)) has exponent dividing m for any a ∈ A #. It is proved that there exist {p, d, c, m}-bounded numbers c 1 and m 1 such that γ c 1 (G) has exponent dividing m 1.  相似文献   

7.
For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ G ) is isomorphic toG × {−1, + 1}.  相似文献   

8.
Group Chromatic Number of Graphs without K5-Minors   总被引:2,自引:0,他引:2  
 Let G be a graph with a fixed orientation and let A be a group. Let F(G,A) denote the set of all functions f: E(G) ↦A. The graph G is A -colorable if for any function fF(G,A), there is a function c: V(G) ↦A such that for every directed e=u vE(G), c(u)−c(v)≠f(e). The group chromatic numberχ1(G) of a graph G is the minimum m such that G is A-colorable for any group A of order at least m under a given orientation D. In [J. Combin. Theory Ser. B, 56 (1992), 165–182], Jaeger et al. proved that if G is a simple planar graph, then χ1(G)≤6. We prove in this paper that if G is a simple graph without a K 5-minor, then χ1(G)≤5. Received: August 18, 1999 Final version received: December 12, 2000  相似文献   

9.
Let H and G be two finite graphs. Define h H (G) to be the number of homomorphisms from H to G. The function h H (·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for A G , the adjacency matrix of a graph G, we have h H (A G ) = h H (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h H (·)1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function h H (·)1/m is a norm.  相似文献   

10.
Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ? H × A, where A is an abelian group. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.  相似文献   

11.
Given an exact category A, we introduce a bisimplicial set W A, which is self-dual and contains both the G-construction of A and its dualization. We prove that the embeddings of G.A and G op .A into W A are homotopy equivalences. If A is an exact category with duality, we calculate the induced action of duality on K 1 (A). We also survey on the self-duality property of some known constructions.  相似文献   

12.
LetG be a finite group, andS a subset ofG \ |1| withS =S −1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S −1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA 5 is a 4-Cl-group, which was a conjecture posed by Li and Praeger.  相似文献   

13.
We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X1, ..., Xm]/(X1l, ..., Xml), G = (\mathbbZ/l\mathbbZ)m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.  相似文献   

14.
Let 𝒜 = (A 1, A 12, A 2) be a locally D 8 amalgam with a finite completion G. Suppose that A 1 ∈ Syl 2(G) and N A 1 (A 12) is Abelian. We determine G/O(G).  相似文献   

15.
Given an open bounded connected subset Ω of ℝn, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.  相似文献   

16.
A proper subgroup A of a finite group G is said to be primitive or meet-irreducible if there is a unique subgroup A0 ≤ G such that A is a maximal subgroup of A0. In this case we say that |A0: A| is the small index of A and denote it by |G: A|0. In this article, we study the influence of meet-irreducible subgroups and their small indexes on the structure of G. In particular, we prove that a finite group G is supersoluble if and only if |G: A|0 = |G: B|0 for any two meet-irreducible subgroups A and B of G with AG = BG.  相似文献   

17.
On Group Chromatic Number of Graphs   总被引:2,自引:0,他引:2  
Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For fF(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function fF(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χg(G). In this note we will prove the following results. (1) Let H1 and H2 be two subgraphs of G such that V(H1)∩V(H2)=∅ and V(H1)∪V(H2)=V(G). Then χg(G)≤min{max{χg(H1), maxvV(H2)deg(v,G)+1},max{χg(H2), maxuV(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K3,3-minor, then χg(G)≤5.  相似文献   

18.
LetG be an arbitrary group with a subgroupA. The subdegrees of (A, G) are the indices [A:AA 9] (wheregεG). Equivalent definitions of that concept are given in [IP] and [K]. IfA is not normal inG and all the subdegrees of (A, G) are finite, we attach to (A, G) the common divisor graph Γ: its vertices are the non-unit subdegrees of (A, G), and two different subdegrees are joined by an edge iff they arenot coprime. It is proved in [IP] that Γ has at most two connected components. Assume that Γ is disconnected. LetD denote the subdegree set of (A, G) and letD 1 be the set of all the subdegrees in the component of Γ containing min(D−{1}). We proved [K, Theorem A] that ifA is stable inG (a property which holds whenA or [G:A] is finite), then the setH={g ε G| [A:AA g ] εD 1 ∪ {1}} is a subgroup ofG. In this case we say thatA<H<G is a disconnected system (briefly: a system). In the current paper we deal with some fundamental types of systems. A systemA<H<G is irreducible if there does not exist 1<N△G such thatAN<H andAN/N<H/N<G/N is a system. Theorem A gives restrictions on the finite nilpotent normal subgroups ofG, whenG possesses an irreducible system. In particular, ifG is finite then Fit(G) is aq-group for a certain primeq. We deal also with general systems. Corollary (4.2) gives information about the structure of a finite groupG which possesses a system. Theorem B says that for any systemA<H<G,N G (N G (A))=N G (A). Theorem C and Corollary C’ generalize a result of Praeger [P, Theorem 2]. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at Tel Aviv University under the supervision of Prof. Marcel Herzog.  相似文献   

19.
Let 𝒜 = (A 1, A 12, A 2) be a locally D 8 amalgam with a finite completion G. Suppose that A 1 ∈ Syl 2(G). We show that under these conditions |A 1| ≤25, or N A 1 (A 12) is Abelian. As applications of our results, we determine all the finite completions G, up to O(G), in the case where N A 1 (A 12) is non-Abelian.  相似文献   

20.
Summary LetA+(k) denote the ring [t]/t k+1 and letG be a reductive complex Lie algebra with exponentsm 1, ...,m n. This paper concerns the Lie algebra cohomology ofGA +(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA +(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m s +1, one has weight 0 and the others have weights (k+1)m s +t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A +(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA n ,B n ,C n , orD n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship  相似文献   

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