共查询到20条相似文献,搜索用时 78 毫秒
1.
N. S. Romanovskii 《Algebra and Logic》2009,48(6):449-464
A soluble group G is said to be rigid if it contains 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 right ℤ[G/G
i
]-modules. Free soluble groups are important examples of rigid groups. A rigid group G is divisible if elements of a quotient G
i
/G
i+1 are divisible by nonzero elements of a ring ℤ[G/G
i
], or, in other words, G
i
/G
i+1 is a vector space over a division ring Q(G/G
i
) of quotients of that ring. A rigid group G is decomposed if it splits into a semidirect product A
1
A
2…A
p
of Abelian groups A
i
≅ G
i
/G
i+1. A decomposed divisible rigid group is uniquely defined by cardinalities α
i
of bases of suitable vector spaces A
i
, and we denote it by M(α1,…, α
p
). The concept of a rigid group appeared in [arXiv:0808.2932v1 [math.GR], ], where the dimension theory is constructed for algebraic geometry over finitely generated rigid groups. In [11], all rigid groups were proved to be equationally Noetherian, and it was stated that any rigid group is embedded in a suitable
decomposed divisible rigid group M(α1,…, α
p
). Our present goal is to derive important information directly about algebraic geometry over M(α1,… α
p
). Namely, irreducible algebraic sets are characterized in the language of coordinate groups of these sets, and we describe
groups that are universally equivalent over M(α1,…, α
p
) using the language of equations. 相似文献
2.
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
3.
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
4.
Edward A. Bertram 《Israel Journal of Mathematics》1991,75(2-3):243-255
We prove first that if G is a finite solvable group of derived length d ≥ 2, then k(G) > |G|1/(2d−1), where k(G) is the number of conjugacy classes in G. Next, a growth assumption on the sequence [G(i): G(i+1)]
1
d−1
, where G(i) is theith derived group, leads to a |G|1/(2d−1) lower bound for k(G), from which we derive a |G|c/log
2log2|G| lower bound, independent of d(G). Finally, “almost logarithmic” lower bounds are found for solvable groups with a nilpotent
maximal subgroup, and for all Frobenius groups, solvable or not. 相似文献
5.
6.
We present the construction for a u-product G1 ○ G2 of two u-groups G1 and G2, and prove that G1 ○ G2 is also a u-group and that every u-group, which contains G1 and G2 as subgroups and is generated by these, is a homomorphic image of G1 ○ G2. It is stated that if G is a u-group then the coordinate group of an affine space Gn is equal to G ○ Fn, where Fn is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian
group or wreath product of two free Abelian groups of finite ranks.
__________
Translated from Algebra i Logika, Vol. 44, No. 5, pp. 601–621, September–October, 2005.
Supported by RFBR grant No. 05-01-00292, by FP “Universities of Russia” grant No. 04.01.053, and by RF Ministry of Education
grant No. E00-1.0-12. 相似文献
7.
B. A. F. Wehrfritz 《Central European Journal of Mathematics》2007,5(4):686-695
Let G be a hypercyclic group. The most substantial results of this paper are the following. a) If G/G′ is 2-divisible, then G is 2-divisible. b) If G/G′ is a 2′-group, then G is a 2′-group. c) If G/G′ is divisible by finite-of-odd-order, then G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O
2′ (G).
相似文献
8.
A. I. Sozutov 《Algebra and Logic》2005,44(6):422-428
An involution i of a group G is said to be perfect in G if any two non-commuting involutions in iG are conjugated by an involution in the same class. We generalize theorems of Jordan and M. Hall concerning sharply doubly
transitive groups, and the Shunkov theorem on periodic groups with a finite isolated subgroup of even order.
__________
Translated from Algebra i Logika, Vol. 44, No. 6, pp. 751–762, November–December, 2005. 相似文献
9.
P. A. Zalesskii 《Monatshefte für Mathematik》2002,135(2):167-171
A profinite group is said to be just infinite if each of its proper quotients is finite. We address the question which profinite
groups admit just infinite quotients. It is proved that any profinite group whose order (as a supernatural number) is divisible
only by finitely many primes admits just infinite quotients. It is shown that if a profinite group G possesses the property in question then so does every open subgroup and every finite extension of G.
Received 20 July 2001 相似文献
10.
S. A. Shakhova 《Algebra and Logic》2006,45(4):277-285
Let ℳ be any quasivariety of Abelian groups, Lq(ℳ) be a subquasivariety lattice of ℳ, dom
G
ℳ
be the dominion of a subgroup H of a group G in ℳ, and G/dom
G
ℳ
(H) be a finitely generated group. It is known that the set L(G, H, ℳ) = {dom
G
N
(H)| N ∈ Lq(ℳ)} forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of dom
G
ℳ
(H). It is proved that the lattice L(G,H,ℳ) is semidistributive and necessary and sufficient conditions are specified for
its being distributive.
__________
Translated from Algebra i Logika, Vol. 45, No. 4, pp. 484–499, July–August, 2006. 相似文献
11.
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
12.
L. Yu. Glebskii 《Mathematical Notes》1999,65(1):31-40
Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg
iBigi
−1 andA+B
i, whereg
i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB
i possess the following property: ‖B
iA−1gBjA−1‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices.
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999. 相似文献
13.
V. D. Mazurov 《Algebra and Logic》2006,45(2):117-123
Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every
two non-commuting members of X generate a subgroup isomorphic to A4 or to A5. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they
generate a subgroup isomorphic to A4. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X)
and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A5 then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer
in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2)
The subgroup 〈XG〉 is a direct product of subgroups 〈C
α〉-generated by some connected components C
α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup
isomorphic to A5. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A5 or is cyclic of order 5.
Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental
Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President)
and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.
__________
Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006. 相似文献
14.
A. I. Sozutov 《Algebra and Logic》2007,46(3):195-199
An involution j of a group G is said to be almost perfect in G if any two involutions in jG whose product has infinite order are conjugated by a suitable involution in jG. Let G contain an almost perfect involution j and |CG(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |CG(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite
index in G.
__________
Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007. 相似文献
15.
A. Borel 《Proceedings Mathematical Sciences》1987,97(1-3):45-52
In this noteG is a locally compact group which is the product of finitely many groups Gs(ks)(s∈S), where ks is a local field of characteristic zero and Gs an absolutely almost simplek
s-group, ofk
s-rank ≥1. We assume that the sum of the rs is ≥2 and fix a Haar measure onG. Then, given a constantc > 0, it is shown that, up to conjugacy,G contains only finitely many irreducible discrete subgroupsL of covolume ≥c (4.2). This generalizes a theorem of H C Wang for real groups. His argument extends to the present case, once it is shown
thatL is finitely presented (2.4) and locally rigid (3.2). 相似文献
16.
WANGWEIFAN ZHANGKEMIN 《高校应用数学学报(英文版)》1997,12(4):455-462
A Planar graph g is called a ipseudo outerplanar graph if there is a subset v.∈V(G),[V.]=i,such that G-V. is an outerplanar graph in particular when G-V.is a forest ,g is called a i-pseudo-tree .in this paper.the following results are proved;(1)the conjecture on the total coloring is true for all 1-pseudo-outerplanar graphs;(2)X1(G) 1 fo any 1-pseudo outerplanar graph g with △(G)≥3,where x4(G)is the total chromatic number of a graph g. 相似文献
17.
0 ,G1,G2,... of supergraphs of G such that Gi is a subgraph of Gj for any i<j and Gi is an optimal (l+i)-edge-connected augmentation of G for any i≥0.
In this paper we will show that the augmentation algorithm of A. Frank [3] can also be used to solve the corresponding Successive
Edge-Augmentation Problem and implies (a stronger version of) the Successive Augmentation Property, even for some non-uniform
demands.
In addition we show the – previously unknown – Successive Augmentation Property for directed edge-connectivity (in the case
of uniform demands).
For several possible extensions and for the two vertex-connectivity versions counter-examples are given.
Received March 1995 / Revised version received February 1997
Published online March 16, 1999 相似文献
18.
Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup
of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups
in a set {U3(2m) | m = 1, 2, …} is isomorphic to U3(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite.
__________
Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008. 相似文献
19.
A commutative Schur ring over a finite group G has dimension at most s G = d 1 + … +d r , where the d i are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2 n ), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant. 相似文献
20.
The Automorphism Group of a Class of Nilpotent Groups with Infinite Cyclic Derived Subgroups 下载免费PDF全文
The automorphism group of a class of nilpotent groups with infinite cyclic derived subgroups is determined. Let G be the direct product of a generalized extraspecial Z-group E and a free abelian group A with rank m, where E ={(1 kα_1 kα_2 ··· kα_nα_(n+1) 0 1 0 ··· 0 α_(n+2)...............000...1 α_(2n+1)000...01|αi∈ Z, i = 1, 2,..., 2 n + 1},where k is a positive integer. Let AutG G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and AutG/ζ G,ζ GG be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G') G→ AutG→ Aut(G')→ 1 is split.(ii) Aut_(G') G/Aut_(G/ζ G,ζ G)G≌Sp(2 n, Z) ×(GL(m, Z)■(Z~)m).(iii) Aut_(G/ζ G,ζ GG/Inn G)≌(Z_k)~(2n)⊕(Z)~(2nm). 相似文献