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1.
We prove a number of facts on metabelian products of metabelian groups, useful in algebraic geometry over groups. Namely, for a metabelian product of arbitrary metabelian groups, we look at the structure of a derived subgroup, and the Fitting radical; find criteria determining when a metabelian product of u-groups is again a u-group; and specify conditions under which a metabelian product of metabelian groups is a strong semidomain.  相似文献   

2.
Let G be a finitely generated torsion-free group, which is a metabelian product of Abelian groups. We establish a criterion for endomorphisms satisfying some extra restriction to be invertible in G. Test sets for IA-endomorphisms of G are described and its test rank computed.  相似文献   

3.
Ushakov  P. V. 《Mathematical Notes》2001,70(5-6):830-837
Let be the free product of two Abelian torsion-free groups, let and , where is the Cartesian subgroup of the group , and let F contain no zero divisors. In the paper it is proved that, in this case, any automorphism of the group is inner. This result generalized the well-known result of Bachmuth, Formanek, and Mochizuki on the automorphisms of groups of the form , , , where is a free group of rank two.  相似文献   

4.
Let qG be a quasivariety generated by a group G and be a non-Abelian quasivariety of groups with a finite lattice of subquasivarieties. Suppose is contained in a quasivariety generated by the following two groups: a free 2-nilpotent group F2( 2) of rank 2 and a free metabelian (i.e., with an Abelian commutant) group F2( 2) of rank 2. It is proved that either = qF2( 2) or = qF2( 2) in this instance.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 389–398, July–August, 2005.  相似文献   

5.
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   

6.
Assume that a quasivariety of groups contains a non-Abelian free metabelian group and a non-Abelian free nilpotent group of class 2. It is proved that the lattice of quasivarieties contained in is infinite and non-modular.  相似文献   

7.
We calculate the test rank of a finite rank free metabelian Lie algebra over an arbitrary field and characterize the test sets for these algebras. We prove that each automorphism that is the identity modulo the derived subalgebra and that acts as the identity on some test set is an inner automorphism.  相似文献   

8.
For the free rank 2 metabelian Lie algebra over an infinite field we prove that an endomorphism of the algebra which preserves the automorphic orbit of a nonzero element is an automorphism. We construct some counterexamples over finite fields.  相似文献   

9.
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.
Pchelintsev  S. V. 《Mathematical Notes》2003,74(1-2):245-254
It is proved that, for any metabelian Mal'tsev algebra M over a field of characteristic 2,3, there is an alternative algebra A such that the algebra M can be embedded in the commutator algebra A(-). Moreover, the enveloping alternative algebra A can be found in the variety of algebras with the identity [x,y][z,t] = 0. The proof of this result is based on the construction of additive bases of the free metabelian Mal'tsev algebra and the free alternative algebra with the identity [x,y][z,t] = 0.  相似文献   

11.
A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ?3 × ?3 k or ?3 × ?3 × ? p where k ≥ 1 and p is a prime. In addition, we prove that ?2 × ?2 × ? p is a Schur group for every prime p.  相似文献   

12.
Krylov  P. A.  Pakhomova  E. G. 《Mathematical Notes》2001,69(3-4):364-372
The structure of the additive group of a regular module is considered. Abelian groups that are regular modules over their rings of endomorphisms are studied. Nonreduced endoregular groups and endoregular torsion-free groups of finite rank are described.  相似文献   

13.
An abelian group is called a mixed one if it is neither torsion nor torsion-free. It is to be proved that every mixed group can be provided with a nonzero associative ring structure. Our methods of proofs are straightforward and elementary.  相似文献   

14.
Nonlinearity of the Automorphism Groups of Some Free Algebras   总被引:2,自引:2,他引:0  
We prove that the groups of tame automorphisms of a free Lie algebra (free associative algebra, absolutely free algebra, algebra of polynomials) of rank at least four over a field of characteristic zero admit no faithful representation by matrices over any field.  相似文献   

15.
Glaz and Wickless introduced the class G of mixed abelian groups A which have finite torsion-free rank and satisfy the following three properties: i) A p is finite for all primes p, ii) A is isomorphic to a pure subgroup of P A P and iii) Hom(A, tA) is torsion. A ring R is a left Kasch ring if every proper right ideal of R has a non-zero left annihilator. We characterize the elements A of G such that E(A)/tE(A) is a left Kasch ring, and discuss related results.  相似文献   

16.
Let F n be the free group of rank n, let Aut(F n ) be its automorphism group and let Out(F n ) be its outer automorphism group. We show that every solvable subgroup of Out(F n ) has a finite index subgroup that is finitely generated and free Abelian. We also show that every Abelian subgroup of Out(F n ) has a finite index subgroup that lifts to Aut(F n ).  相似文献   

17.
18.
This work is devoted to the study of properties of verbal products and interlacings of groups. More precisely we study approximation properties (for example, the finite approximability, approximability by finite p-groups, and so on) of nilpotent and metabelian products and interlacings. We apply verbal interlacings to the problem of finite approximability with respect to the conjugacy for some nilpotent interlacings and to the study of stable automorphisms of groups of the form $ {F \left/ {V} \right.}\left( \mathcal{N} \right) $ .  相似文献   

19.
Consider the group scheme where R is an arbitrary commutative ring with 1 0 and a unitx R* acts on R by multiplication. We will study the finiteness properties of subgroups of G(OS)where OS is an S-arithmetic subring of a global function field.The subgroups we are interested in are of the form where Q is a subgroup of OS*. The finiteness propertiesof these metabelian groups can be expressed in terms of the-invariant due to R. Bieri and R. Strebel. Theorem A. Let S be a finite set of places of a global functionfield (regarded as normalized discrete valuations) and OS thecorresponding S-arithmetic ring. Let Q be a subgroup of OS*.Then Q is finitely generated and for all integers n 1 the followingare equivalent:
(1) OS Q is of type FPn;
(2) OS is n-tameas a ZQ-module;
(3) each p S restricts to a non-trivial homomorphism and the set is n-tame.
If these conditions hold for at least one n 1 then the identity holds.} Theorem B. Let r denote the rank of Q. Then the followinghold:
(1) the group OS Q is not of type FPr+1};
(2) if Qhas maximum rank r = |S| –1 then the group OS Q is oftype FPr.
In particular, is of type FP|S| –1 but not of type FP|S|. 1991 Mathematics SubjectClassification: 20E08, 20F16, 20G30, 52A20.  相似文献   

20.
A reflexive topological group G is called strongly reflexive if each closed subgroup and each Hausdorff quotient of the group G and of its dual group is reflexive. In this paper we establish an adequate concept of strong reflexivity for convergence groups. We prove that complete metrizable nuclear groups and products of countably many locally compact topological groups are BB-strongly reflexive.  相似文献   

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