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1.
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
2.
V. M. Kopytov 《Algebra and Logic》1998,37(3):170-180
We construct an example of a fully orderable group that is not locally solvable. It is also shown that a free group is embedded
in a fully orderable group. To meet these ends, use is made of a group of invertible formal power series with zero free term
under composition.
Supported by RFFR grant No. 96-01-00088.
Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 301–319, May–June, 1998. 相似文献
3.
We argue that for any subgroup H of rank 1 in a multiplicative group of positive reals, among Dlab groups of the closed intervalI=[0],[1] on an extended set
of reals, there exist groups DH*(I) and DH* which lack normal relatively convex subgroups, are not simple groups, and have just two distinct linear orders. The cardinality
of a set of linear orders on Dlab groups is computed. It is established that every rigid l-group is Abelian if it belongs
to a varietyD of l-groups groups generated by the linearly ordered groups DH*(I) and DH*. We prove that the quasivariety q(DH*(I), DH*) of groups generated by DH*(I) and DH* is distinct from a quasivarietyO of all orderable groups. Similar results are stated for a variety of l-groups and the quasivariety of groups that are not
embeddable in DH*(I) and DH*.
Supported by RFFR grant No. 96-01-00088.
Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 531–548, September–October, 1999. 相似文献
4.
V. D. Mazurov 《Algebra and Logic》2000,39(1):42-49
The article contains two characterizations of projective linear groups PGL2(P) over a locally finite field P of characteristic 2: the first is defined in terms of permutation groups, and the second,
in terms of a structure of involution centralizers. One of the two is used to prove the existence of infinite groups which
are recognizable by the set of their element orders.
In memory of Viktor A. Gorbunov
Supported by RFFR grant No. 99-01-00550.
Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 74–86, January–February, 2000. 相似文献
5.
V. V. Bludov 《Algebra and Logic》2005,44(6):370-380
We give examples of linearly ordered groups that are not embeddable in divisible orderable. In the first example, the group
does not embed in any divisible group with strictly isolated unity. In the second example, the group in question is an O*-group,
and in the third, it is a group with a central system of convex subgroups.
To my teacher A. I. Kokorin
Supported by RFBR grant Nos. 96-01-00358, 99-01-00335, and 03-01-00320.
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Translated from Algebra i Logika, Vol. 44, No. 6, pp. 664–681, November–December, 2005. 相似文献
6.
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
7.
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
8.
For any natural k ≥ 3 and l ≥ 2, we describe decidability boundaries for two varieties: the variety of all k-nilpotent groups
and the variety of all l-solvable groups.
Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 127–133, March–April, 2000. 相似文献
9.
V. V. Bludov 《Algebra and Logic》2003,42(5):304-317
We prove a theorem saying that in finitely generated linearly ordered metabelian groups there exists a finite system of normal convex subgroups satisfying orderability conditions for groups, and an embedding theorem for linearly ordered metabelian groups whose initial linear orders extend to -divisible linearly ordered metabelian ones. As a consequence, it is stated that orderable metabelian groups are embedded, with extension of all their linear orders, in -divisible orderable metabelian groups. 相似文献
10.
Maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups,
are treated. For every such torus, its expression as a direct sum of cyclic groups is obtained in an explicit form.
Supported by RFBR grant Nos. 05-01-00797 and 06-01-39001, and by SB RAS Integration Project No. 2006.1.2.
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Translated from Algebra i Logika, Vol. 46, No. 2, pp. 129–156, March–April, 2007. 相似文献
11.
It is proved that finite simple groups L4(2m), m ⩾ 2, and U4(2m), m ⩾ 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups.
As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order
8.
Supported by RFBR (grant Nos. 05-01-00797 and 06-01-39001), by SB RAS (Complex Integration project No. 1.2), and by the Ministry
of Education of China (Project for Retaining Foreign Expert).
Supported by NSF of Chongqing (CSTC: 2005BB8096).
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Translated from Algebra i Logika, Vol. 47, No. 1, pp. 83–93, January–February, 2008. 相似文献
12.
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
13.
N. S. Romanovskii 《Algebra and Logic》2007,46(4):274-280
The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases:
(1) G = Fn is a free metabelian group of rank n; (2) G = Wn,k is a wreath product of free Abelian groups of ranks n and k.
Supported by RFBR grant No. 05-01-00292.
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Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007. 相似文献
14.
V. D. Mazurov 《Algebra and Logic》1997,36(3):182-192
It is proved that the permutation wreath product H of a simple Suzuki group Sz(27) and a subgroup fo a symmetric group of degree 23, isomorphic to a Frobenius group of order 253, is (up to isomorphism) distinguished
among all finite groups by the set of orders of its elements. Since H possesses a minimal normal subgroup N that contains
an element of order equal to the exponent of N, this result furnishes a counterexample to one of the conjectures set forth
by Shi [1]. In addition, we show that the direct square of a group Sz(27) is also distinguished by the set of orders of its elements.
Supported by RFFR grant No. 96-01-01893.
Translated fromAlgebra i Logika, Vol. 36, No. 3, pp. 304–322, May–June, 1997. 相似文献
15.
It is proved that a finite group isomorphic to a simple non-Abelian group L3(2m) or U3(2m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S4(2m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all
recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1.
Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by
the State Education Ministry of China (grant No. 98083).
Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000. 相似文献
16.
It is proved that if L is one of the simple groups 3D4(q) or F4(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is
isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group.
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Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005.
Supported by RFBR grant No. 04-01-00463. 相似文献
17.
18.
V. D. Mazurov 《Algebra and Logic》2000,39(3):189-198
We prove that a group which contains elements of orders 1, 2, 3, 4, 5 and does not contain elements of any other order is
locally finite and isomorphic either to an alternating group of degree 6 or to an extension of a nontrivial elementary Abelian
2-group by an alternating group of degree 5.
This article was written during my visit to the University of Manitoba, Canada, and supported by RFFR grant No. 99-01-00550.
Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 329–346, May–June, 2000. 相似文献
19.
N. N. Kushpel 《Journal of Mathematical Sciences》2006,136(3):3957-3965
A linear group G ≤ GL(V) is called same-invariant if the subspace of linear invariants Vg is one and the same for all g ∈ G, g ≠ 1. In this paper, we consider finite same-invariant linear groups of orders pq, (p,
q) = 1, or p2 over a field of characteristic p. Bibliography: 8 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 224–239. 相似文献
20.
For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime
graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set
with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders
on these sets; the information obtained is collected in tables. We consider several applications of these results to various
problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups.
Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science
Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher
School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant
No. 86-197.
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Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005. 相似文献