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1.
For a pro-p groupG, containing a free pro-p open normal subgroup of rank at most 2, a characterization as the fundamental group of a connected graph of cyclic groups
of order at mostp, and an explicit list of all such groups with trivial center are given. It is shown that any automorphism of a free pro-p group of rank 2 of coprime finite order is induced by an automorphism of the Frattini factor groupF/F
*
. Finally, a complete list of automorphisms of finite order, up to conjugacy in Aut(F), is given.
Supported by an NSERC grant.
Supported by the Austrian Science Foundation. 相似文献
2.
M. V. Neshadim 《Algebra and Logic》1996,35(5):316-318
An automorphism of an arbitrary group is called normal if all subgroups of this group are left invariant by it. Lubotski [1]
and Lue [2] showed that every normal automorphism of a noncyclic free group is inner. Here we prove that every normal automorphism
of a nontrivial free product of groups is inner as well.
Supported by RFFR grant No. 13-011-1513.
Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 562–566, September–October, 1996. 相似文献
3.
A Dehn twist automorphism of a group G is an automorphism which can be given (as specified below) in terms of a graph-of-groups decomposition of G with infinite cyclic edge groups. The classic example is that of an automorphism of the fundamental group of a surface which
is induced by a Dehn twist homeomorphism of the surface. For , a non-abelian free group of finite rank n, a normal form for Dehn twist is developed, and it is shown that this can be used to solve the conjugacy problem for Dehn
twist automorphisms of .
Received: February 12, 1996. 相似文献
4.
Finite groups of Lie type form the greater part of known finite simple groups. An important class of subgroups of finite groups
of Lie type are so-called reductive subgroups of maximal rank. These arise naturally as Levi factors of parabolic groups and
as centralizers of semisimple elements, and also as subgroups with maximal tori. Moreover, reductive groups of maximal rank
play an important part in inductive studies of subgroup structure of finite groups of Lie type. Yet a number of vital questions
dealing in the internal structure of such subgroups are still not settled. In particular, we know which quasisimple groups
may appear as central multipliers in the semisimple part of any reductive group of maximal rank, but we do not know how normalizers
of those quasisimple groups are structured. The present paper is devoted to tackling this problem.
Supported by RFBR (grant No. 05-01-00797) and by SB RAS (Young Researchers Support grant No. 29 and Integration project No.
2006.1.2).
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Translated from Algebra i Logika, Vol. 47, No. 1, pp. 3–30, January–February, 2008. 相似文献
5.
M. A. Shevelin 《Siberian Mathematical Journal》2012,53(5):934-942
We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field. 相似文献
6.
We examine the palindromic automorphism group , of a free group F
n
, a group first defined by Collins in [5] which is related to hyperelliptic involutions of mapping class groups, congruence
subgroups of , and symmetric automorphism groups of free groups. Cohomological properties of the group are explored by looking at a contractible
space on which acts properly with finite quotient. Our results answer some conjectures of Collins and provide a few striking results about
the cohomology of , such as that its rational cohomology is zero at the vcd.
Received: January 17, 2000. 相似文献
7.
S. A. Zyubin 《Algebra and Logic》2006,45(5):296-305
A subgroup having non-empty intersection with each class of conjugate elements of the group is said to be conjugately dense.
It is shown that, under certain conditions, the number of conjugately dense subgroups in a free product with amalgamation
is not less than some cardinal. As a consequence, P. Neumann’s conjecture in the Kourovka notebook (Question 6.38) is refuted.
It is also stated that a modular group and a non-Abelian group of countable or finite rank possess continuum many pairwise
non-conjugate conjugately dense subgroups.
Supported by RFBR grant No. 03-01-00905.
__________
Translated from Algebra i Logika, Vol. 45, No. 5, pp. 520–537, September–October, 2006. 相似文献
8.
M. I. Gordin 《Journal of Mathematical Sciences》1997,87(6):4067-4071
The homoclinic group (an invariant with respect to topological conjugacy) for hyperbolic toral automorphisms is determined.
Certain conditions are given for conjugacy of a homeomorphism of a compact space to hyperbolic toral automorphism. Bibliography:
7 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1994, pp. 140–147.
This paper is partially supported by Russian Foundation for Basic Research, grant 94-01-00921.
Translated by V. V. Sadovskaya. 相似文献
9.
G. A. Bazhenova 《Algebra and Logic》2000,39(4):215-223
We deal with a class of rational subsets of a group, that is, the least class of its subsets which contains all finite subsets
and is closed under taking union. a product of two sets, and under generating of a submonoid by a set. It is proved that the
class of rational subsets of a finitely generated nilpotent group G is a Boolean algebra iff G is Abelian-by-finite. We also
study the question asking under which conditions the set of solutions for equations in groups will be rational. It is shown
that the set of solutions for an arbitrary equation in one variable in a finitely generated nilpotent group of class 2 is
rational. And we give an example of an equation in one variable in a free nilpotent group of nilpotency class 3 and rank 2
whose set of solutions is not rational.
Supported by RFFR grant No. 98-01-00932.
Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 379–394, July–August, 2000. 相似文献
10.
E. P. Vdovin 《Algebra and Logic》2007,46(2):90-119
In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it
is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together
with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups
of every finite group are conjugate.
Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) for Support of Young Russian Scientists
via projects MK-1455.2005.1 and MK-3036.2007.1; by SB RAS Young Researchers Support grant No. 29; via Integration Project
No. 2006.1.2.
__________
Translated from Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007. 相似文献
11.
Let P be a locally finite group of prime exponent p. We prove that if P admits a finite soluble automorphism group G of order n coprime to p, such that the fixed point group C
P(G)is soluble of derived length d, then P is nilpotent of class bounded by a function of p, n, and d. A similar statement is shown to hold for Lie (p - 1)-Engel algebras; it is analogous to the Bergman-Isaacs theorem proved for associative rings, provided the condition of being soluble for an automorphism group is added. Our proof is based on a generalization of Kreknin's theorem concerning the solubility of Lie rings with a regular automorphism of finite order. This generalization, giving an affirmative answer to a question of Winter and extending one of his results to the case of infinitedimensional Lie algebras, is interesting in its own right. Moreover, we use a generalization of Higgins' theorem on the nilpotency of soluble Lie Engel algebras.
Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 706-723, November-December, 1995.Supported by RFFR grant No. 94-01-00048-a and by ISF grant NQ7000. 相似文献
12.
Seonhee Lim 《Journal of Pure and Applied Algebra》2008,212(7):1632-1663
We develop an explicit covering theory for complexes of groups, parallel to that developed for graphs of groups by Bass. Given a covering of developable complexes of groups, we construct the induced monomorphism of fundamental groups and isometry of universal covers. We characterize faithful complexes of groups and prove a conjugacy theorem for groups acting freely on polyhedral complexes. We also define an equivalence relation on coverings of complexes of groups, which allows us to construct a bijection between such equivalence classes, and subgroups or overgroups of a fixed lattice Γ in the automorphism group of a locally finite polyhedral complex X. 相似文献
13.
E. P. Vdovin 《Algebra and Logic》2000,39(5):301-312
Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved
that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|2<|G|.
Supported through FP “Integration” project No. 274, by RFFR grant No. 99-01-00550, by International Soros Education Program
for Exact Sciences (ISEP) grant No. S99-56, and by a SO RAN grant for Young Scientists, Presidium Decree No. 83 of 03/10/2000.
Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 526–546, September—October, 2000. 相似文献
14.
V. A. Roman’kov N. G. Khisamiev A. A. Konyrkhanova 《Siberian Mathematical Journal》2017,58(3):536-545
We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT n (Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT n (Z) has no proper existentially closed subgroups. 相似文献
15.
An automorphism of a (profinite) group is called normal if each (closed) normal subgroup is left invariant by it. An automorphism
of an abstract group is p-normal if each normal subgroup of p-power, where p is prime, is left invariant. Obviously, the inner
automorphism of a group will be normal and p-normal. For some groups, the converse was stated to be likewise true. N. Romanovskii
and V. Boluts, for instance, established that for free solvable pro-p-groups of derived length 2, there exist normal automorphisms
that are not inner. Let N2 be the variety of nilpotent groups of class 2 and A the variety of Abelian groups. We prove the following results: (1) If
p is a prime number distinct from 2, then the normal automorphism of a free pro-p-group of rank ≥2 in N2A is inner (Theorem 1); (2) If p is a prime number distinct from 2, then the p-normal automorphism of an abstract free N2A-group of rank ≥2 is inner (Theorem 2).
Supported by RFFR grant No. 93-01-01508.
Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 249–267, May–June, 1996. 相似文献
16.
A. N. Starkov 《Mathematical Notes》1998,63(2):233-241
A criterion for a wide class of topological groups which includes linear discrete groups and Lie groups to be Tychonoff groups
is established. The main result provides a criterion for an almost polycyclic group to have the Tychonoff property. By the
well-known Tits alternative, this yields the required criterion for linear discrete groups. In conclusion it is pointed out
that a particular case of the presented proof yields a Tychonoff property criterion for Lie groups. In addition, an example
of a polycyclic group without Tychonoff subgroups of finite index is constructed.
Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 269–279, February, 1998.
The author wishes to express his gratitude to R. I. Grigorchuk for setting the problem and his interest in the work.
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00182 and by the American Mathematical
Society Fund. 相似文献
17.
We construct a full class of nilpotent groups of class 2 of an arbitrary infinite
cardinality . Their centers, commutator subgroups and factors modulo the center will be the
same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will
coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ . 相似文献
18.
A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular
Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex.
Supported in part by the N.Z. Marsden Fund (grant no. UOA0124). 相似文献
19.