Let G be a graph and denote by Q(G)=D(G) A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given. 相似文献
LetK be a field,G a finite group.G is calledK-admissible iff there exists a finite dimensionalK-central division algebraD which is a crossed product forG. Now letK andL be two finite extensions of the rationalsQ such that for every finite groupG, G isK-admissible if and only ifG isL-admissible. ThenK andL have the same degree and the same normal closure overQ.
An erratum to this article is available at . 相似文献
Let
t: D ?D¢\tau: {\cal D} \rightarrow{\cal D}^\prime
be an equivariant holomorphic map of symmetric domains associated to a homomorphism
r: \Bbb G ?\Bbb G¢{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime
of semisimple algebraic groups defined over
\Bbb Q{\Bbb Q}
. If
G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q})
and
G¢ ì \Bbb G¢(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q})
are torsion-free arithmetic subgroups with
r (G) ì G¢{\bf\rho} (\Gamma) \subset \Gamma^\prime
, the map G\D ?G¢\D¢\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime
of arithmetic varieties and the rationality of D{\cal D}
and
D¢{\cal D}^\prime
as well as the commensurability groups of
s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C})
determines a conjugate equivariant holomorphic map
ts: Ds ?D¢s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma}
of fs: (G\D)s ?(G¢\D¢)s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma
of . We prove that is rational if is rational. 相似文献
Zusammenfassung LetD:G→GL(n,C) be an irreducible linear representation of a finite groupG with the characterX. IfD is realizible in Q(ξm) and Q(ξm′) we give a condition for then realizability ofD in Q(ξ(m′)). If the degreen is a prime ≠ 2, we show thatD realizible in Q(ξf), wheref is the conductor of the abelian extensionQ(X)/Q. 相似文献
Let G be a finite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (|Q|, |H|) = 1 and (|H|, |QG|) ≠ 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H ≦ HτG, where HτG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let ℱ be a saturated formation containing all supersoluble groups and let X ≦ E be normal subgroups of a group G such that G/E ∈ ℱ. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F* (E), then G ∈ ℱ. 相似文献
We have described the structure of Qn(G) = Δn(G)/Δn+1(G) for 35 particular classes of groups G with order 25 in the previous article. In this article, the structure of Qn(G) for all the remaining classes of groups G with order 25 are presented. 相似文献
In this article we present the nth power Δn(G) of the augmentation ideal Δ(G) and describe the structure of Qn(G) = Δn(G)/Δn+1(G) for 35 particular groups G of order 25. The structure of Qn(G) for all the remaining groups of order 25 will be determined in a forthcoming article. 相似文献
The author studies a DG-module A such that D is a Dedekind domain, A/CA(G) is not an Artinian D-module, CA(G) = 1, G is a soluble group, and the system of all subgroups H ≤ G for which the quotient modules A/CA(H) are not Artinian D-modules satisfies the minimum condition. The structure of G is described. 相似文献
Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are c-normal in G. 相似文献
According to a classical result of Burnside, if G is a finite 2-group, then the Frattini subgroup Φ(G) of G cannot be a nonabelian group of order 8. Here we study the next possible case, where G is a finite 2-group and Φ(G) is nonabelian of order 16. We show that in that case Φ(G) ≅ M × C2, where M ≅ D8 or M ≅ Q8 and we shall classify all such groups G (Theorem A).
Received: 16 February 2005; revised: 7 March 2005 相似文献
Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/NrdD/F(D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK1(D) = 1 and F*2 = F*2n. Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated 相似文献
In this article a class of subgroups of a finite group G, called Q-injectors, is introduced. If G is soluble, the Q-injectors are precisely the injectors of the Fitting sets. A characterization of nilpotent Q-injectors is given as well as a sufficient condition for the solubility of a finite group G, in terms of Q-injectors, which generalizes a well known result. 相似文献
Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup, and X a compact manifold (or more generally, a Borel space) on which G acts. Assume that ν is a μ -stationary measure on X, where μ is an admissible measure on G, and that the G-action is essentially free. We consider the foliation of K\ X with Riemmanian leaves isometric to the symmetric space K\ G, and the associated tangential bounded de-Rham cohomology, which we show is an invariant of the action. We prove both vanishing
and nonvanishing results for bounded tangential cohomology, whose range is dictated by the size of the maximal projective
factor G/Q of (X, ν). We give examples showing that the results are often best possible. For the proofs we formulate a bounded tangential version
of Stokes’ theorem, and establish a bounded tangential version of Poincaré’s Lemma. These results are made possible by the
structure theory of semisimple Lie groups actions with stationary measure developed in Nevo and Zimmer [Ann of Math. 156, 565--594]. The structure theory assert, in particular, that the G-action is orbit equivalent to an action of a uniquely determined parabolic subgroup Q. The existence of Q allows us to establish Stokes’ and Poincaré’s Lemmas, and we show that it is the size of Q (determined by the entropy) which controls the bounded tangential cohomology.
Supported by BSF and ISF.
Supported by BSF and NSF. 相似文献
ABSTRACT Let D be a finite dimensional F -central division algebra and G an irreducible subgroup of D*: = GL1(D). Here we investigate the structure of D under various group identities on G. In particular, it is shown that when [D:F] = p2, p a prime, then D is cyclic if and only if D* contains a nonabelian subgroup satisfying a group identity. 相似文献
The holomorph of a group G is NormB(λ(G)), the normalizer of the left regular representation λ(G) in its group of permutations B = Perm(G). The multiple holomorph of G is the normalizer of the holomorph in B. The multiple holomorph and its quotient by the holomorph encodes a great deal of information about the holomorph itself and about the group λ(G) and its conjugates within the holomorph. We explore the multiple holomorphs of the dihedral groups Dn and quaternionic (dicyclic) groups Qn for n ≥ 3. 相似文献
Let k be an algebraically closed base field of arbitrary characteristic. In this paper, we study actions of a connected solvable
linear algebraic group G on a central simple algebra Q. The main result is the following: Q can be split G-equivariantly by a finite-dimensional splitting field, provided that G acts “algebraically,” i.e., provided that Q contains a G-stable order on which the action is rational. As an application, it is shown that rational torus actions on prime PI-algebras
are induced by actions on commutative domains.
Presented by Paul Smith. 相似文献
LetR*G be a crossed product of the groupG over the prime ringR and assume thatR*G is also prime. In this paper we study unitsq in the Martindale ring of quotientsQ0(R*G) which normalize bothR and the group of trivial units ofR*G. We obtain quite detailed information on their structure. We then study the group ofX-inner automorphisms ofR*G induced by such elements. We show in fact that this group is fairly close to the group of automorphisms ofR*G induced by certain trivial units inQ0(R)*G. As an application we specialize to the case whereR=U(L) is the enveloping algebra of a Lie algebraL. Here we study the semi-invariants forL andG which are contained inQ0(R*G) and we obtain results which extend known properties ofU(L). Finally, every cocommutative Hopf algebraH over an algebraically closed field of characteristic 0 is of the formH=U(L)*G. Thus we also obtain information on the semi-invariants forH contained inQ0(H).
Research supported in part by N.S.F. Grant Nos. MCS 83-01393 and MCS 82-19678. 相似文献
If G is a group of odd order and χ ∈ Irr(G) lifts an irreducible Brauer character ?, then we associate to χ a canonical pair (Q, δ) up to G-conjugacy, where Q is a vertex of ? and δ ∈ Irr(Q) is a linear character of Q. We show that (Q, δ) is a Navarro vertex for χ. We also discuss examples. 相似文献
It is well known that the smallest eigenvalue of the adjacency matrix of a connected d-regular graph is at least ? d and is strictly greater than ? d if the graph is not bipartite. More generally, for any connected graph G = (V, E), consider the matrix Q = D + A where D is the diagonal matrix of degrees in the graph G and A is the adjacency matrix of G. Then Q is positive semidefinite, and the smallest eigenvalue of Q is 0 if and only if G is bipartite. We will study the separation of this eigenvalue from 0 in terms of the following measure of nonbipartiteness of G. For any S ? V, we denote by emin(S) the minimum number of edges that need to be removed from the induced subgraph on S to make it bipartite. Also, we denote by cut(S) the set of edges with one end in S and the other in V ? S. We define the parameter Ψ as. The parameter Ψ is a measure of the nonbipartiteness of the graph G. We will show that the smallest eigenvalue of Q is bounded above and below by functions of Ψ. For d-regular graphs, this characterizes the separation of the smallest eigenvalue of the adjacency matrix from ?d. These results can be easily extended to weighted graphs. 相似文献