scholarly journals INVARIANTS FOR METABELIAN GROUPS OF PRIME POWER EXPONENT, COLORINGS AND STAIRS

2021 ◽  
pp. 1-29
Author(s):  
JONATHAN ARIEL BARMAK
Keyword(s):  
Prime Power ◽  
Power Exponent ◽  
Metabelian Groups

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 .

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

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 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.

scholarly journals Kostrikin’s theorem on Engel groups of prime power exponent

1968 ◽  
Vol 26 (2) ◽  
pp. 197-213 ◽  
Author(s):  
Seymour Bachmuth ◽  
Horace Mochizuki
Keyword(s):  
Prime Power ◽  
Power Exponent

UPPER BOUNDS IN THE RESTRICTED BURNSIDE PROBLEM II

1996 ◽  
Vol 06 (06) ◽  
pp. 735-744 ◽  
Author(s):  
MICHAEL VAUGHAN-LEE ◽  
E.I. ZELMANOV
Keyword(s):  
Finite Groups ◽  
Prime Power ◽  
Upper Bounds ◽  
Reduction Theorem ◽  
Power Exponent ◽  
Burnside Problem ◽  
Restricted Burnside Problem

We show that if G is a finite m generator group of exponent n, with m>1, then [Formula: see text] This result extends bounds previously obtained for finite groups of prime power exponent. The proof is based on a reduction theorem for the restricted Burnside problem due to Hall and Higman.

Burnside metabelian groups

1968 ◽  
Vol 307 (1490) ◽  
pp. 235-250 ◽  
Keyword(s):  
Direct Product ◽  
Prime Power ◽  
Lower Central Series ◽  
Nilpotency Class ◽  
General Situation ◽  
Power Exponent ◽  
Central Series ◽  
Augmentation Ideal ◽  
Cyclic Groups ◽  
Number Of Generators

If B is a group of prime-power exponent p e and solubility class 2, then B has nilpotency class at most e ( p e — p e-1 )+1 provided the number of generators of B are at most p +1. Representa­tions of B are constructed which in the case of two generators and prime exponent is a faithful representation of the free group of the variety under study and for prime-power exponent show the existence of a group with nilpotency class e ( p e — p e-1 ). In the general situation where B as above has exponent n , and n is not a prime-power, the place where the lower central series of G becomes stationary is determined by a knowledge of the nilpotency class of the groups of prime-power exponent for all prime divisors of n . The bound e ( p e — p e-1 )+1 on the nilpotency class is a consequence of the following: Let G be a direct product of at most p —1 cyclic groups of order p e and R the group ring of G over the integers modulo p e . Then the e ( p e — p e-1 ) th power of the augmentation ideal of R is contained in the ideal of R generated by all 'cyclotomic’ polynomials Ʃ p e -1 i = 0 g i for all g in G . If G is a direct product of more than p +1 cyclic groups, then this result is no longer true unless e = 1.

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