Bounds on the Engel class of some group extensions

1972 ◽  
Vol 71 (2) ◽  
pp. 179-188 ◽  
Author(s):  
David Shield
Keyword(s):  
Normal Subgroup ◽  
Prime Power ◽  
Upper Bounds ◽  
Nilpotency Class ◽  
Power Exponent ◽  
Group Extensions

AbstractIn this paper upper bounds on the Engel length of a group G of prime-power exponent are obtained, under varying conditions on a normal subgroup H and the quotient G/H. Examples are constructed to show that the bounds are best possible when the nilpotency class of H is less than the relevant prime.

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.

scholarly journals Landau’s theorem on conjugacy classes for normal subgroups

2016 ◽  
Vol 26 (07) ◽  
pp. 1453-1466
Author(s):  
Antonio Beltrán ◽  
María José Felipe ◽  
Carmen Melchor
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Upper Bounds ◽  
Conjugacy Classes ◽  
Index Formula ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
Small Index ◽  
Positive Integers

Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly [Formula: see text] conjugacy classes for any positive integer [Formula: see text]. We show that, for any positive integers [Formula: see text] and [Formula: see text], there exist finitely many finite groups [Formula: see text], up to isomorphism, having a normal subgroup [Formula: see text] of index [Formula: see text] which contains exactly [Formula: see text] non-central [Formula: see text]-conjugacy classes. Upper bounds for the orders of [Formula: see text] and [Formula: see text] are obtained; we use these bounds to classify all finite groups with normal subgroups having a small index and few [Formula: see text]-classes. We also study the related problems when we consider only the set of [Formula: see text]-classes of prime-power order elements contained in a normal subgroup.

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 The class of a nilpotent wreath product

1977 ◽  
Vol 17 (1) ◽  
pp. 53-89 ◽  
Author(s):  
David Shield
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Group Ring ◽  
Upper Bound ◽  
Lower Central Series ◽  
Nilpotency Class ◽  
Central Series ◽  
Augmentation Ideal ◽  
Cambridge Philos ◽  
Base Group

Let G be a group with a normal subgroup H whose index is a power of a prime p, and which is nilpotent with exponent a power of p. Gilbert Baumslag (Proc. Cambridge Philos. Soc. 55 (1959), 224–231) has shown that such a group is nilpotent; the main result of this paper is an upper bound on its nilpotency class in terms of parameters of H and G/H. It is shown that this bound is attained whenever G is a wreath product and H its base group.A descending central series, here called the cpp-series, is involved in these calculations more closely than is the lower central series, and the class of the wreath product in terms of this series is also found.Two tools used to obtain the main result, namely a useful basis for a finite p-group and a result about the augmentation ideal of the integer group ring of a finite p-group, may have some independent interest. The main result is applied to the construction of some two-generator groups of large nilpotency class with exponents 8, 9, and 25.

scholarly journals On c-normality of finite groups

2005 ◽  
Vol 78 (3) ◽  
pp. 297-304 ◽  
Author(s):  
M. Asaad ◽  
M. Ezzat Mohamed
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order

AbstractA subgroup H of a finite G is said to be c-normal in G if there exists a normal subgroup N of G such that G = HN with H ∩ N ≤ HG = CoreG(H). We are interested in studying the influence of the c–normality of certain subgroups of prime power order on the structure of finite 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 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

Supplement to “The group E6(q) and graphs with a locally linear group of automorphisms” by V. I. Trofimov and R. M. Weiss

2009 ◽  
Vol 148 (1) ◽  
pp. 33-45 ◽  
Author(s):  
V. I. TROFIMOV
Keyword(s):  
Normal Subgroup ◽  
Vector Space ◽  
Permutation Group ◽  
Linear Group ◽  
Prime Power ◽  
Companion Paper ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Group Of Automorphisms ◽  
Locally Linear

AbstractLet q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.

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