scholarly journals On the modular isomorphism problem for groups of class 3 and obelisks

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Leo Margolis ◽  
Mima Stanojkovski
Keyword(s):  
Group Algebra ◽  
Lower Central Series ◽  
Isomorphism Problem ◽  
Nilpotency Class ◽  
Positive Answer ◽  
Central Series ◽  
New Techniques ◽  
New Approach ◽  
Modular Group Algebra ◽  
Recent Computer

Abstract We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.

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.

Lie Dimension Subgroups and Central Series Related to Group Algebras

2009 ◽  
Vol 16 (03) ◽  
pp. 427-436
Author(s):  
Ernesto Spinelli
Keyword(s):  
Group Algebra ◽  
Positive Characteristic ◽  
Nilpotency Class ◽  
Group Algebras ◽  
Central Series ◽  
Characteristic P

Let KG be the group algebra of a group G over a field K of positive characteristic p, and let 𝔇(n)(G) and 𝔇[n](G) denote the n-th upper Lie dimension subgroup and the n-th lower one, respectively. In [1] and [12], the equality 𝔇(n)(G) =𝔇[n](G) is verified when p ≥ 5. Motivated by [16, Problem 55], in the present paper we establish it for particular classes of groups when p ≤ 3. Finally, we introduce and study a new central series of G linked with the Lie nilpotency class of KG.

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.

The Characterisation of Modular Group Algebras Having Unit Groups of Nilpotency Class 3

1995 ◽  
Vol 38 (1) ◽  
pp. 112-116 ◽  
Author(s):  
M. Anwar Rao ◽  
Robert Sandling
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Unit Group ◽  
Nilpotency Class ◽  
Group Algebras ◽  
Modular Group Algebra

AbstractThe unit group of the modular group algebra of a finite p-group in characteristic p is nilpotent. The p-groups for which it is of nilpotency class 3 were determined in work of Coleman, Passman, Shalev and Mann when p ≥ 3. We resolve the p = 2 case here which completes the classification.

On nilpotency of the group of outer class-preserving automorphisms of a group

2015 ◽  
Vol 15 (02) ◽  
pp. 1650026 ◽  
Author(s):  
Pradeep K. Rai
Keyword(s):  
Lower Central Series ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Central Series ◽  
Trivial Group ◽  
Improve Bound

Let G be a group and Zj(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Out c(G/Zj(G)), the group of outer class-preserving automorphisms of G/Zj(G), is nilpotent of class k, then Out c(G) is nilpotent of class at most j + k. Moreover, if Out c(G/Zj(G)) is a trivial group, then Out c(G) is nilpotent of class at most j. As an application we prove that if γi(G)/γi(G) ∩ Zj(G) is cyclic then Out c(G) is nilpotent of class at most i + j, where γi(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Out c(G) for some classes of nilpotent groups G.

scholarly journals The modular group algebra problem for groups of order p5

1996 ◽  
Vol 61 (2) ◽  
pp. 229-237 ◽  
Author(s):  
Mohamed A. M. Salim ◽  
Robert Sandling
Keyword(s):  
Group Algebra ◽  
Modular Group ◽  
Nilpotency Class ◽  
Group Algebras ◽  
Modular Group Algebra

AbstractWe show that p-groups of order p5 are determined by their group algebras over the field of p elements. Many cases have been dealt with in earlier work of ourselves and others. The only case whose details remain to be given here is that of groups of nilpotency class 3 for p odd.

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