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1.
In the following,G denotes a finite group,r(G) the number of conjugacy classes ofG, β(G) the number of minimal normal subgroups ofG andα(G) the number of conjugate classes ofG not contained in the socleS(G). Let Φ
j
= {G|β(G) =r(G) −j}. In this paper, the family Φ11 is classified. In addition, from a simple inspection of the groups withr(G) =b conjugate classes that appear in ϒ
j
=1/11
Φ
j
, we obtain all finite groups satisfying one of the following conditions: (1)r(G) = 12; (2)r(G) = 13 andβ(G) > 1; …; (9)r(G) = 20 andβ(G) > 8; (10)r(G) =n andβ(G) =n −a with 1 ≦a ≦ 11, for each integern ≧ 21. Also, we obtain all finite groupsG with 13 ≦r(G) ≦ 20,β(G) ≦r(G) − 12, and satisfying one of the following conditions: (i) 0 ≦α(G) ≦ 4; (ii) 5 ≦α(G) ≦ 10 andS(G) solvable. 相似文献
2.
Aner Shalev 《Israel Journal of Mathematics》1994,87(1-3):153-160
LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude
that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC
G(a) for 1≠a ∈A. 相似文献
3.
Wojciech Jaworski 《Journal d'Analyse Mathématique》1996,68(1):183-208
Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL
∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure
on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary. 相似文献
4.
Dorothee Schueth 《Commentarii Mathematici Helvetici》1995,70(1):434-454
It is known that ifH
m is the classical (2m+1)-dimensional Heisenberg group, Γ a cocompact discrete subgroup ofH
m andg a left invariant metric, then (Γ/H
m, g) is infinitesimally spectrally rigid within the family of left invariant metrics. The purpose of this paper is to show that
for everym≥2 and for a certain choice of Γ andg, there is a deformation (Γ/H
m, g
α) withg=g
1 such that for every α≠1, (Γ/H
m, g
α)does admit a nontrivial isospectral deformation. For α≠1 the metricsg
α will not beH
m-left invariant, and the (Γ/H
m, gxα) will not be nilmanifolds, but still solvmanifolds. 相似文献
5.
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
6.
S. S. Dragomir 《Journal of Applied Mathematics and Computing》2000,7(3):611-627
An Ostrowski type integral inequality for the Riemann-Stieltjes integral ∫ab ƒ (t) du (t), where ƒ is assumed to be of bounded variation on [a,b] andu is ofr-H- H?lder type on the same interval, is given. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out. 相似文献
7.
Michael L. Mihalik 《Commentarii Mathematici Helvetici》1996,71(1):362-372
If a finitely presented groupG is negatively curved, automatic or asynchronously automatic thenG has an asynchronously bounded “almost prefix closed” combing. Results in [Br1] and [E] imply that the fundamental group of any closed 3-manifold satisfying Thurston's geometrization conjecture has an
asynchronously bounded, almost prefix closed combing.
MAIN THEOREM. IfM is a compactP
2-irreducible 3-manifold,π
1 (M) has an asynchronously bounded, almost prefix closed combing, andH, a subgroup ofπ
1 (M), is quasiconvex with respect to this combing, then the cover ofM corresponding toH is a missing boundary manifold. 相似文献
8.
G. L. O’brien 《Israel Journal of Mathematics》1981,39(1-2):145-154
LetG be a finite directed graph which is irreducible and aperiodic. Assume each vertex ofG leads to at least two other vertices, and assumeG has a cycle of prime length which is a proper subset ofG. Then there exist two functionsr:G →G andb:G →G such that ifr(x)=y andb(x)=z thenx →y andx →z inG andy ≠z and such that some composition ofr’s andb’s is a constant function.
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. I am grateful to Cornell
University whose kind hospitality I enjoyed while working on this problem. 相似文献
9.
A group is said to have finite (special) rank ≤ sif all of its finitely generated subgroups can be generated byselements. LetGbe a locally finite group and suppose thatH/HGhas finite rank for all subgroupsHofG, whereHGdenotes the normal core ofHinG. We prove that thenGhas an abelian normal subgroup whose quotient is of finite rank (Theorem 5). If, in addition, there is a finite numberrbounding all of the ranks ofH/HG, thenGhas an abelian subgroup whose quotient is of finite rank bounded in terms ofronly (Theorem 4). These results are based on analogous theorems on locally finitep-groups, in which case the groupGis also abelian-by-finite (Theorems 2 and 3). 相似文献
10.
Thomas Meixner 《Israel Journal of Mathematics》1985,51(1-2):68-78
For a (finite) groupG and some prime powerp
n, theH
p
n
-subgroupH
pn (G) is defined byH
p
n
(G)=〈xεG|x
pn≠1〉. A groupH≠1 is called aH
p
n
-group, if there is a finite groupG such thatH is isomorphic toH
p
n
(G) andH
p
n
(G)≠G. It is known that the Fitting length of a solvableH
p
n
-group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH
p
n
-groups having Fitting lengthn, this result is “almost” best possible. 相似文献
11.
In [2] we proved that ifG is a finite group containing an involution whose centralizer has order bounded by some numberm, thenG contains a nilpotent subgroup of class at most two and index bounded in terms ofm. One of the steps in the proof of that result was to show that ifG is soluble, then ¦G/F(G) ¦ is bounded by a function ofm, where F (G) is the Fitting subgroup ofG. We now show that, in this part of the argument, the involution can be replaced by an arbitrary element of prime order. 相似文献
12.
Dan Segal 《Israel Journal of Mathematics》1996,94(1):7-19
A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│
a
subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ). 相似文献
13.
Henri Moscovici 《Israel Journal of Mathematics》1973,15(3):230-236
LetG be a Lie group,H a closed subgroup,L a unitary representation ofH andU
L the corresponding induced representation onG. The main result of this paper, extending Ol’ŝanskii’s version of the Frobenius reciprocity theorem, expresses the intertwining
number ofU
L and an irreducible unitary representationV ofG in terms ofL and the restriction ofV
∞ toH. 相似文献
14.
Claus Scheiderer 《manuscripta mathematica》1996,91(1):247-281
LetG be a profinite group which has an open subgroupH such that the cohomologicalp-dimensiond≔cdp(H) is finite (p is a fixed prime). The main result of this paper expresses thep-primary part of high degree cohomology ofG in terms of the elementary abelianp-subgroups ofG: From the latter one constructs a natural profinite simplicial setA
G, on whichG acts by conjugation. ThenH
n(G,M)≅H
G
n
(AG,M) holds forn≧d+r and everyp-primary discreteG-moduleM (r≔p-rank ofG). If one uses profinite Farrell cohomology, which is introduced in this paper, the analogous fact holds in all degrees. These
results are the profinite analogues of theorems by K.S. Brown for discrete groups. 相似文献
15.
Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F
1,F
2,⃛,F
1| ofG, if |E(H)∩E(F
1)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G).
Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences. 相似文献
16.
LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)⩽(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively.
Project supported by the National Natural Science Foundation of China. 相似文献
17.
LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI
3(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I
2(G)I(H)+I(G)I(H)I(G)+I(H)I2(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH
2(G/H′, T)≤1, are computed. the subgroup ofG determined byI
n(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained 相似文献
18.
LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev.
This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars
(No. 10425103) and the Key Program of NSF (No. 10331020) in China. 相似文献
19.
Leila Schneps 《Israel Journal of Mathematics》1996,93(1):125-144
LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL
1 of ℚ(T) with Galois groupG. LetL be the subfield ofL
1 fixed byH. We make the hypothesis thatL
1 admits a quadratic extensionL
2 which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL
1 then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL
2. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL
1 admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions
of ℚ(T). 相似文献
20.
We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair
of groups (H;G; α; β) is deformed using a combinatorial datum (σ; v; r) consisting of an automorphism σ of H, a permutation v of the set G and a transition map r: G → H in order to obtain a new matched pair (H; (G; *); α′, β′) such that there exists a σ-invariant isomorphism of groups H
α⋈β
G ≅H
α′⋈β′
(G, *). Moreover, if we fix the group H and the automorphism σ ∈ Aut H then any σ-invariant isomorphism H
α⋈β
G ≅ H
α′⋈β′
G′ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications
two Schreier type classification theorems for bicrossed products of groups are given. 相似文献