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1.
2.
Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed
than it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an
orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group
G of length three with periodic G/G′ and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed.
To the memory of N. Ya. Medvedev
Supported by RFBR (project No. 03-01-00320).
Translated from Algebra i Logika, Vol. 48, No. 3, pp. 291–308, May–June, 2009. 相似文献
3.
Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L
of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic
to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that
are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established
for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of
dimension at most 26.
Supported by RFBR (project Nos. 08-01-00322 and 06-01-39001), by SB RAS (Integration Project No. 2006.1.2), and by the Council
for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and Young Doctors and Candidates
of Science (grants MD-2848.2007.1 and MK-377.2008.1).
Translated from Algebra i Logika, Vol. 47, No. 5, pp. 558–570, September–October, 2008. 相似文献
4.
P. M. Akhmet’ev 《Journal of Mathematical Sciences》2009,159(6):753-760
In this paper, a geometric approach to stable homotopy groups of spheres based on the Pontryagin–Thom construction is proposed.
From this approach, a new proof of the Hopf-invariant-one theorem of J. F. Adams for all dimensions except 15, 31, 63, and
127 is obtained. It is proved that for n > 127, in the stable homotopy group of spheres Π
n
, there is no element with Hopf invariant one. The new proof is based on geometric topology methods. The Pontryagin–Thom theorem
(in the form proposed by R. Wells) about the representation of stable homotopy groups of the real, projective, infinite-dimensional
space (these groups are mapped onto 2-components of stable homotopy groups of spheres by the Kahn–Priddy theorem) by cobordism
classes of immersions of codimension 1 of closed manifolds (generally speaking, nonoriented) is considered. The Hopf invariant
is expressed as a characteristic class of the dihedral group for the self-intersection manifold of an immersed codimension-1
manifold that represents the given element in the stable homotopy group. In the new proof, the geometric control principle
(by M. Gromov) for immersions in the given regular homotopy classes based on the Smale–Hirsch immersion theorem is required.
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 3–15, 2007. 相似文献
5.
We consider one–factorizations of complete graphs which possess an automorphism group fixing k ≥ 0 vertices and acting regularly (i.e., sharply transitively) on the others. Since the cases k = 0 and k = 1 are well known in literature, we study the case k≥ 2 in some detail. We prove that both k and the order of the group are even and the group necessarily contains k − 1 involutions. Constructions for some classes of groups are given. In particular we extend the result of [7]: let G be an abelian group of even order and with k − 1 involutions, a one–factorization of a complete graph admitting G as an automorphism group fixing k vertices and acting regularly on the others can be constructed. 相似文献
6.
Let G be a reductive group over an algebraically closed field k. Consider the moduli space of stable principal G-bundles on a smooth projective curve C over k. We give necessary and sufficient conditions for the existence of Poincaré bundles over open subsets of this moduli space,
and compute the orders of the corresponding obstruction classes. This generalizes the previous results of Newstead, Ramanan
and Balaji–Biswas–Nagaraj–Newstead to all reductive groups, to all topological types of bundles, and also to all characteristics. 相似文献
7.
Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A1 ⩾ A2 ⩾ … An ⩾ … such that ⋂An = 1 and all factors An/An+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that
the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary
derived lengths and ranks are equationally Noetherian.
Supported by RFBR grant No. 05-01-00292.
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Translated from Algebra i Logika, Vol. 46, No. 1, pp. 46–59, January–February, 2007. 相似文献
8.
V. A. Churkin 《Siberian Mathematical Journal》2010,51(3):557-568
The weak Bieberbach theorem states that each crystallographic group on a Euclidean space uniquely determines its translation
lattice as an abstract group. Garipov proved in 2003 that the same holds for crystallographic groups on Minkowski spaces and
asked whether a similar claim holds in the pseudo-Euclidean spaces ℝ
p,q
. We prove that the weak Bieberbach theorem holds for crystallographic groups on pseudo-Euclidean spaces ℝ
p,q
with min{p, q} ≤ 2. For min{p, q} ≥ 3 we construct examples of crystallographic groups with two distinct lattices exchanged by a suitable automorphism of
the group. For crystallographic groups with two distinct isomorphic pseudo-Euclidean lattices we also prove that the coranks
of their intersection in these lattices can take arbitrary values greater than 2 with the exception of 4. 相似文献
9.
We study groups having the big powers property BP. It is proved that if a pregroup satisfies some natural axioms, then its
universal group has this property. In particular, fundamental groups of some graphs of groups have the big powers property
if BP holds for edge and vertex subgroups and a number of natural conditions are satisfied. The results obtained are applied
to Lyndon’s completions U(P)ℤ[t] of the universal group U(P) with P satisfying the conditions mentioned.
Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday
Translated from Algebra i Logika, Vol. 48, No. 3, pp. 342–377, May–June, 2009. 相似文献
10.
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.
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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
11.
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.
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Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008. 相似文献
12.
A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be
seminormal in a group G if there exists a subgroup B such that G = AB and AB1 is a proper subgroup of G, for every proper subgroup B1 of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite
group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the
classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements
imposed on the groups is unnecessary.
Supported by BelFBR grant Nos. F05-341 and F06MS-017.
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Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007. 相似文献
13.
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.
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Translated from Algebra i Logika, Vol. 44, No. 6, pp. 751–762, November–December, 2005. 相似文献
14.
Non-nilpotent, finitely generated, associative nil-algebras are studied as well as their adjoint groups and Golod groups.
Solutions are given to some problems in residually finite group theory, questions posed in the Kourovka Notebook included.
Supported by RFBR grant No. 03-01-00356.
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Translated from Algebra i Logika, Vol. 45, No. 2, pp. 231–238, March–April, 2006. 相似文献
15.
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
16.
A number of conditions are specified which are sufficient for totally ordered groups with polycyclic factor group to contain
a finite normal series of convex subgroups whose factors possess good enough properties. In any case studying such totally
ordered groups is reduced to treating extensions of these groups as well as their virtually o-equivalent extensions. The concept
of a virtually o-equivalent extension is a particular case of the notion of an Archimedean extension.
Supported by RFBR project No. 03-01-00320.
Translated from Algebra i Logika, Vol. 47, No. 5, pp. 529–540, September–October, 2008. 相似文献
17.
Jin Ho KWAK Ju Mok OH 《数学学报(英文版)》2006,22(5):1305-1320
A graph G is one-regular if its automorphism group Aut(G) acts transitively and semiregularly on the arc set. A Cayley graph Cay(Г, S) is normal if Г is a normal subgroup of the full automorphism group of Cay(Г, S). Xu, M. Y., Xu, J. (Southeast Asian Bulletin of Math., 25, 355-363 (2001)) classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. (Croat. Chemica Acta, 73, 969-981 (2000)) classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut(G) are cyclic. A classification of the same kind of graphs of valency 6 is also discussed. 相似文献
18.
O. Yu. Dashkova 《Algebra and Logic》2008,47(5):340-347
We are concerned with locally soluble linear groups of infinite central dimension and infinite sectional p-rank, p ⩾ 0, in
which every proper non-Abelian subgroup of infinite sectional p-rank has finite central dimension. It is proved that such
groups are soluble.
Translated from Algebra i Logika, Vol. 47, No. 5, pp. 601–616, September–October, 2008. 相似文献
19.
Spectra of finite linear and unitary groups 总被引:1,自引:0,他引:1
A. A. Buturlakin 《Algebra and Logic》2008,47(2):91-99
The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural
number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special
linear and unitary groups.
Supported by RFBR (grant Nos. 08-01-00322 and 06-01-39001) and by the Council for Grants (under RF President) and State Aid
of Leading Scientific Schools (project NSh-344.2008.1).
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Translated from Algebra i Logika, Vol. 47, No. 2, pp. 157–173, March–April, 2008. 相似文献
20.
E. I. Timoshenko 《Algebra and Logic》2006,45(1):67-74
For a factor group with respect to periodic part of a group of the form F/[R′, F], an embedding in the matrix group is defined.
The criteria for a matrix to belong to an image of this group and for elements to be conjugate are specified. Some statements
having a direct bearing on groups of the form in question are proved. Application of the results obtained allows us to refine
the answer in [7] to a question by O. Chapuis concerning the universal classification of ∀-free soluble groups with two generators.
Supported by RFBR grant No. 02-01-00293 and by FP “Universities of Russia” grant No. UR.04.01.227.
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Translated from Algebra i Logika, Vol. 45, No. 1, pp. 114–125, January–February, 2006. 相似文献