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1.
A group is said to be p-rigid, where p is a natural number, if it has a normal series of the form G = G
1 > G
2 > … > G
p
> G
p+1 = 1, whose quotients G
i
/G
i+1 are Abelian and are torsion free when treated as
\mathbbZ \mathbb{Z} [G/G
i
]-modules. Examples of rigid groups are free soluble groups. We point out a recursive system of universal axioms distinguishing
p-rigid groups in the class of p-soluble groups. It is proved that if F is a free p-soluble group, G is an arbitrary p-rigid group, and W is an iterated wreath product of p infinite cyclic groups, then ∀-theories for these groups satisfy the inclusions A(F) ê A(G) ê A(W) \mathcal{A}(F) \supseteq \mathcal{A}(G) \supseteq \mathcal{A}(W) . We construct an ∃-axiom distinguishing among p-rigid groups those that are universally equivalent to W. An arbitrary p-rigid group embeds in a divisible decomposed p-rigid group M = M(α1,…, α
p
). The latter group factors into a semidirect product of Abelian groups A
1
A
2…A
p
, in which case every quotient M
i
/M
i+1 of its rigid series is isomorphic to A
i
and is a divisible module of rank αi over a ring
\mathbbZ \mathbb{Z} [M/M
i
]. We specify a recursive system of axioms distinguishing among M-groups those that are Muniversally equivalent to M. As a consequence, it is stated that the universal theory of M with constants in M is decidable. By contrast, the universal theory of W with constants is undecidable. 相似文献
2.
N. S. Romanovskii 《Algebra and Logic》2008,47(6):426-434
A soluble group G is rigid if it contains a normal series of the form G = G1 > G2 > … > Gp > Gp+1 = 1, whose quotients Gi/Gi+1 are Abelian and are torsion-free as right ℤ[G/Gi]-modules. The concept of a rigid group appeared in studying algebraic geometry over groups that are close to free soluble.
In the class of all rigid groups, we distinguish divisible groups the elements of whose quotients Gi/Gi+1 are divisible by any elements of respective groups rings Z[G/Gi]. It is reasonable to suppose that algebraic geometry over divisible rigid groups is rather well structured. Abstract properties
of such groups are investigated. It is proved that in every divisible rigid group H that contains G as a subgroup, there is
a minimal divisible subgroup including G, which we call a divisible closure of G in H. Among divisible closures of G are divisible
completions of G that are distinguished by some natural condition. It is shown that a divisible completion is defined uniquely
up to G-isomorphism.
Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1).
Translated from Algebra i Logika, Vol. 47, No. 6, pp. 762–776, November–December, 2008. 相似文献
3.
N. S. Romanovskii 《Algebra and Logic》2009,48(2):147-160
A group G is said to be rigid if it contains a normal series of the form G = G
1 > G
2 > … > G
m
> G
m + 1 = 1, whose quotients G
i
/G
i + 1 are Abelian and are torsion free as right Z[G/G
i
]-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group
uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any n, every system of equations in x
1, …, x
n
over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being
Noetherian on G
n
, which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed.
It is proved that every rigid group is equationally Noetherian.
Supported by RFBR (project No. 09-01-00099) and by the Russian Ministry of Education through the Analytical Departmental Target
Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1.419).
Translated from Algebra i Logika, Vol. 48, No. 2, pp. 258–279, March–April, 2009. 相似文献
4.
Edoardo Ballico 《Annali dell'Universita di Ferrara》1985,31(1):63-70
Summary Fixr≥2,N=r(r+3)/2 andN smooth plane curvesA
1…,A
N with degA
i>-2 fori=l,…,N. Then the monodromy group for the plane curves of degreer tangent tog
iAi, gi∈PGL(3), is the full symmetric group.
Supported in part by NATO junior fellowship at M.I.T. 相似文献
Riassunto Sianor≥2,N=r(r+3)/2 eA 1…,A N curve piane lisce di grado almeno 2. Si dimostra che la monodromia per le curve piane di grador tangenti ag 1 A i,g i∈PGL(3), è il gruppo simmetrico.
Supported in part by NATO junior fellowship at M.I.T. 相似文献
5.
OD-characterization of Almost Simple Groups Related to U3(5) 总被引:1,自引:0,他引:1
Let G be a finite group with order |G|=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg- Kegel graph) denoted .by г(G) (or GK(G)). This graph is constructed as follows: The vertex set of it is π(G) = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge (and we write pi - pj) if and only if G contains an element of order pipj. The degree deg(pi) of a vertex pj ∈π(G) is the number of edges incident on pi. We define D(G) := (deg(p1), deg(p2),..., deg(pk)), which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that |H| = |G| and D(H) = D(G). Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L := U3(5) be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable. 相似文献
6.
Let G = GL
N
or SL
N
as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2
N
: Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H
*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H
i
(G, M) are Noetherian A
G
-modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product
of Grassmannians. 相似文献
7.
Samit Dasgupta Gyula Károlyi Oriol Serra Balázs Szegedy 《Israel Journal of Mathematics》2001,126(1):17-28
LetA={a
1, …,a
k} andB={b
1, …,b
k} be two subsets of an Abelian groupG, k≤|G|. Snevily conjectured that, whenG is of odd order, there is a permutationπ ∈S
ksuch that the sums α
i
+b
i
, 1≤i≤k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even whenA is a sequence ofk<|G| elements, i.e., by allowing repeated elements inA. In this last sense the result does not hold for other Abelian groups. With a new kind of application of the polynomial method
in various finite and infinite fields we extend Alon’s result to the groups (ℤ
p
)
a
and
in the casek<p, and verify Snevily’s conjecture for every cyclic group of odd order.
Supported by Hungarian research grants OTKA F030822 and T029759.
Supported by the Catalan Research Council under grant 1998SGR00119.
Partially supported by the Hungarian Research Foundation (OTKA), grant no. T029132. 相似文献
8.
A. Schrijver 《Discrete and Computational Geometry》1991,6(1):527-574
In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ2, a subset {I
1, …,I
p} of the faces ofG, and pathsC
1, …,C
k inG, with endpoints on the boundary ofI
1 ∪ … ∪I
p; find: pairwise disjoint simple pathsP
1, …,P
k inG so that, for eachi=1, …,k, P
i is homotopic toC
i in the space ℝ2\(I
1 ∪ … ∪I
p).
Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm
to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ2 and pairwise disjoint setsW
1, …,W
k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT
1, …,T
k ofG whereT
i
(i=1, …, k). 相似文献
9.
LetA={a
1, …,a
k} and {b
1, …,b
k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata
i+b
i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to
the groupsZ
p
r
andZ
p
rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that
Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|.
This work has been supported partly by NSFC grant number 19971058 and 10271080. 相似文献
10.
George Purdy 《Israel Journal of Mathematics》1978,30(1-2):54-56
If ann-dimensional polytope has facets of areaA
1,A
2, …,A
m, then 2A
i <A
1+…+A
m fori=1,…,m. We show here that conversely these inequalities also ensure the existence of a polytope having these areas. 相似文献
11.
For finite sets of integers A
1,…,A
n
we study the cardinality of the n-fold sumset A
1+…+ A
n
compared to those of (n−1)-fold sumsets A
1+…+A
i−1+A
i+1+…+A
n
. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition
of elements is restricted to an addition graph between the sets. 相似文献
12.
In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M
n+1(p) of order p
n+1 and exponent p
n
for
n \geqslant 2{n \geqslant 2}, and where A is not necessarily a full defect p-block but its defect group P = M
n+1(p) is normal in G. The proof is independent of the classification of finite simple groups. 相似文献
13.
Highest weight representations of a Lie algebra of Block type 总被引:2,自引:0,他引:2
Yue-zhu WU & Yu-cai SU Department of Mathematics Shanghai Jiaotong University Shanghai China Department of Mathematics Qufu Normal University Qufu China Department of Mathematics University of Science Technology of China Hefei China 《中国科学A辑(英文版)》2007,50(4):549-560
For a field F of characteristic zero and an additive subgroup G of F, a Lie algebra B(G) of the Block type is defined with the basis {Lα,i, c|α∈G, -1≤i∈Z} and the relations [Lα,i,Lβ,j] = ((i 1)β- (j 1)α)Lα β,i j αδα,-βδi j,-2c,[c, Lα,i] = 0. Given a total order (?) on G compatible with its group structure, and anyα∈B(G)0*, a Verma B(G)-module M(A, (?)) is defined, and the irreducibility of M(A,(?)) is completely determined. Furthermore, it is proved that an irreducible highest weight B(Z )-module is quasifinite if and only if it is a proper quotient of a Verma module. 相似文献
14.
For an additive subgroup G of a field F of characteristic zero, a Lie algebra B(G) of Block type is defined with basis {Lα,i| α∈G, i∈Z+} and relations [Lα,i, Lβ,j] = (β-α)Lα+β,i+j+(αj-βi)Lα+β,Lα+β,i+j-1.It is proved that an irreducible highest weight B(Z)-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B(G)0^*(the dual space of B(G)0 = span{L0,i|i∈Z+}), a Verma B(G)-module M(∧,λ) is defined, and the irreducibility of M(A,λ) is completely determined. 相似文献
15.
San Ling 《Israel Journal of Mathematics》2000,116(1):117-123
For an abelian varietyA over ℚ
p
, the special fibre in the Néron model ofA over ℤ
p
is the extension of a finite group scheme over ℤ
p
, called the group of connected components, by the connected component of identity. WhenA is the Jacobian variety of an algebraic curve, its component group has been calculated in many cases. We determine in this
paper the component group of thep-new subvariety ofJ
0(M
p
), forM>1 a positive integer andp≥5 a prime not dividingM. Such a subvariety is not the Jacobian of any obvious curve, but it is not clear if it can ever be realised as the Jacobian
of a curve. 相似文献
16.
Let T2k+1 be the set of trees on 2k+1 vertices with nearly perfect matchings and α(T) be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k+1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T(1) and T(12) in T2k+1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T(1)and T^′12,T^″12∈T(12) and each i,j with 2≤i〈j≤11,α(T^′1)=α(T^″1)〉α(Tj)〉α(T^′12)=α(T^″12).It is also shown that for each tree T with T∈T2k+1/(T(1)∪{T2,T3,…,T11}∪T(12)),α(T^′12)〉α(T). 相似文献
17.
N. S. Romanovskii 《Algebra and Logic》2011,49(6):539-550
Let ε = (ε
1, . . . , ε
m
) be a tuple consisting of zeros and ones. Suppose that a group G has a normal series of the form G = G
1 ≥ G
2 ≥ . . . ≥ G
m
≥ G
m+1 = 1, in which G
i > G
i+1 for ε
i = 1, G
i = G
i+1 for ε
i
= 0, and all factors G
i
/G
i+1 of the series are Abelian and are torsion free as right ℤ[G/G
i
]-modules. Such a series, if it exists, is defined by the group G and by the tuple ε uniquely. We call G with the specified series a rigid m-graded group with grading ε. In a free solvable group of derived length m, the above-formulated condition is satisfied by a series of derived subgroups. We define the concept of a morphism of rigid
m-graded groups. It is proved that the category of rigid m-graded groups contains coproducts, and we show how to construct a coproduct G◦H of two given rigid m-graded groups. Also it is stated that if G is a rigid m-graded group with grading (1, 1, . . . , 1), and F is a free solvable group of derived length m with basis {x
1, . . . , x
n
}, then G◦F is the coordinate group of an affine space G
n
in variables x
1, . . . , x
n
and this space is irreducible in the Zariski topology. 相似文献
18.
We say that n independent trajectories ξ1(t),…,ξ
n
(t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ
i
(t
i
) and ξ
j
(t
j
) is at least ɛ, for some indices i, j and for all large enough t
1,…,t
n
, with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition
function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by
the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct
−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ(α) are asymptotically separated, where ξ(α) is the α-process generated by −(−Δ)α/2.
Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000
RID="*"
ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782
RID="**"
ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573 相似文献
19.
John Rizkallah 《代数通讯》2017,45(4):1785-1792
20.
Let (GA)
n
[k](a), A
n
(a), G
n
(a) be the third symmetric mean of k degree, the arithmetic and geometric means of a
1, …, a
n
(a
i
> 0, i = 1, …, n), respectively. By means of descending dimension method, we prove that the maximum of p is k−1/n−1 and the minimum of q is n/n−1(k−1/k)
k/n
so that the inequalities {fx505-1} hold. 相似文献