On metabelian groups of prime-power exponent

1969 ◽  
Vol 310 (1502) ◽  
pp. 393-399 ◽  
Keyword(s):  
Prime Power ◽  
Metabelian Group ◽  
Power Exponent ◽  
Metabelian Groups ◽  
The Right

Gupta, Newman & Tobin (1968) show that in a metabelian group of exponent dividing p k , the subgroup generated by p k -1 th powers is nilpotent. In this paper we obtain the ‘right’ bound for the class of this subgroup together with some information about the subgroup generated by p h th powers, thus answering a question raised by Gupta et al .

On metabelian groups of prime-power exponent

1968 ◽  
Vol 302 (1469) ◽  
pp. 237-242 ◽  
Keyword(s):  
Prime Power ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Power Exponent ◽  
Metabelian Groups

Meier-Wunderli has shown that every metabelian group of exponent p is nilpotent. Here we show that the subgroup generated by p k-1 th powers of elements in a metabelian group of exponent p k is nilpotent. We also obtain some information on the subgroup generated by p k-2 th powers. Finally we obtain a bound for the nilpotency class of n -generator metabelian groups of exponent p k .

scholarly journals On varieties of metabelian groups of prime-power exponent

1972 ◽  
Vol 14 (2) ◽  
pp. 129-154 ◽  
Author(s):  
M. S. Brooks
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Product Variety ◽  
Power Exponent ◽  
Metabelian Groups ◽  
Proper Subvariety

Let Un denote the variety of abelian groups of exponent dividing n, and let p be an arbitrary prime. In this paper all non-nilpotent, join-ireducible subvarieties of the product variety UpUp2 are determined. The proper subvarieties of this kind in fact form an infinite ascending chain …, and an arbitrary proper subvariety B of UpUp2 is either nilpotent or a join , where L is nilpotent and k is uniquely determined by B.

Metabelian groups and varieties

1969 ◽  
Vol 1 (1) ◽  
pp. 15-25 ◽  
Author(s):  
R. A. Bryce
Keyword(s):  
Prime Power ◽  
Power Exponent ◽  
Metabelian Groups ◽  
Universal Algebras

Here we announce results which will appear in full in due course elsewhere. The aim is a study of varieties of metabelian groups. The technique is to study varieties of certain universal algebras, which are very similar to groups, called bigroups. For certain varieties of bigroups all the non-nilpotent join-irreducible subvarieties are determined, and this is used to reduce the same problem for varieties of metabelian groups to the case of prime-power exponent. Questions of distributivity of the lattice of metabelian varieties are also discussed.

On 2-generator metabelian groups of prime-power exponent

1981 ◽  
Vol 37 (1) ◽  
pp. 385-400 ◽  
Author(s):  
Rex S. Dark ◽  
Martin L. Newell
Keyword(s):  
Prime Power ◽  
Power Exponent ◽  
Metabelian Groups

scholarly journals Certain locally nilpotent varieties of groups

2003 ◽  
Vol 67 (1) ◽  
pp. 115-119
Author(s):  
Alireza Abdollahi
Keyword(s):  
Prime Power ◽  
Power Exponent ◽  
Variety Of Groups ◽  
Periodic Groups ◽  
Locally Nilpotent

Let c ≥ 0, d ≥ 2 be integers and be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that is locally nilpotent? We prove that if c ≤ 2d + 2d−1 − 3 then the variety is locally nilpotent and we reduce the question of Gupta about the periodic groups in to the prime power exponent groups in this variety.

Groupings of Metabelian Groups and Extension Categories

1980 ◽  
Vol 32 (2) ◽  
pp. 449-459 ◽  
Author(s):  
K. W. Roggenkamp
Keyword(s):  
Group Ring ◽  
General Result ◽  
Prime Divisor ◽  
Characteristic Zero ◽  
Integral Domain ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Exact Sequences

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall proveTHEOREM. If RG ≌ RH as R-algebras, then G ≌ HThe question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules

scholarly journals Metabelian groups with the same finite quotients

1974 ◽  
Vol 11 (1) ◽  
pp. 115-120 ◽  
Author(s):  
P.F. Pickel
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Isomorphism Classes ◽  
Split Extensions ◽  
Finite Quotients ◽  
Finitely Presented

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. Two groups G and H are said to have the same finite quotients if F(G) = F(H). We construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring. These groups also give an example of infinitely many nonisomorphic split extensions of a fixed finitely presented metabelian group by a fixed finite abelian group, all having the same finite quotients.

A note on the cohomology of metabelian groups

1985 ◽  
Vol 98 (3) ◽  
pp. 437-445 ◽  
Author(s):  
P. H. Kropholler
Keyword(s):  
Finite Rank ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.

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