Some finite groups with zero deficiency

1974 ◽  
Vol 18 (1) ◽  
pp. 73-75 ◽  
Author(s):  
J. W. Wamsley
Keyword(s):  
Finite Groups ◽  
Generators And Relations ◽  
Number Of Generators

AbstractWe introduce further finite groups which can be presented with an equal number of generators and relations.

A Comment on Finite Nilpotent Groups of Deficiency Zero

1980 ◽  
Vol 23 (3) ◽  
pp. 313-316 ◽  
Author(s):  
Edmund F. Robertson
Keyword(s):  
Finite Group ◽  
Finite Number ◽  
Nilpotent Groups ◽  
Metacyclic Groups ◽  
Generators And Relations ◽  
Number Of Generators ◽  
Nilpotent Class ◽  
A Finite Group

A finite group is said to have deficiency zero if it can be presented with an equal number of generators and relations. Finite metacyclic groups of deficiency zero have been classified, see [1] or [6]. Finite non-metacyclic groups of deficiency zero, which we denote by FD0-groups, are relatively scarce. In [3] I. D. Macdonald introduced a class of nilpotent FD0-groups all having nilpotent class≤8. The largest nilpotent class known for a Macdonald group is 7 [4]. Only a finite number of nilpotent FD0-groups, other than the Macdonald groups, seem to be known [5], [7]. In this note we exhibit a class of FD0-groups which contains nilpotent groups of arbitrarily large nilpotent class.

scholarly journals An algorithm for analysis of the structure of finitely presented Lie algebras

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Vladimir V. Kornyak
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nonlinear Partial Differential Equations ◽  
Practical Importance ◽  
Physical Models ◽  
Generators And Relations ◽  
Defining Relations ◽  
Number Of Generators ◽  
Finite Set ◽  
Finitely Presented

International audience We consider the following problem: what is the most general Lie algebra satisfying a given set of Lie polynomial equations? The presentation of Lie algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance, covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are constructionof prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie algebras arising in different physical models. The finite presentations also indicate a way to q-quantize Lie algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason, in practice one needs to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie algebra and its commutator table, and its implementation in the C language. Some computer results illustrating our algorithmand its actual implementation are also presented.

MINIMAL NUMBER OF GENERATORS AND COMPOSITION LENGTH OF SOCLE IN FINITE GROUPS

1996 ◽  
Vol 47 (2) ◽  
pp. 157-163 ◽  
Author(s):  
FRANCESCA DALLA VOLTA ◽  
ANDREA LUCCHINI
Keyword(s):  
Finite Groups ◽  
Number Of Generators

Inequalities for Baer Invariants of Finite Groups

1998 ◽  
Vol 41 (4) ◽  
pp. 385-391 ◽  
Author(s):  
John Burns ◽  
Graham Ellis
Keyword(s):  
Exact Sequence ◽  
Finite Groups ◽  
Generating Sets ◽  
Number Of Generators ◽  
Explicit Bound

AbstractIn this note we further our investigation of Baer invariants of groups by obtaining, as consequences of an exact sequence of A. S.-T. Lue, some numerical inequalities for their orders, exponents, and generating sets. An interesting group theoretic corollary is an explicit bound for |γc+1 (G)| given that G, Zc(G) is a finite p-group with prescribed order and number of generators.

scholarly journals On a class of finite groups

1975 ◽  
Vol 19 (3) ◽  
pp. 290-291
Author(s):  
J. W. Wamsley
Keyword(s):  
Finite Groups ◽  
Normal Closure ◽  
Number Of Generators ◽  
Image Position

Let G be a finite P-group. Denote dim H1 (G, Zp) by d(G) and dimH2(G, Zp) by r(G), then d(G) is the minimal number of generators of G and G has a presentation where F is free on x1, …, xd(G) and R is the normal closure in F of R1, …, Rm. We have always that m ≧ r(G) = d(R/[F, R]) and we say that G belongs to a class, Gp, of the finite pgroups if m = r(G). It is well known (see for example Johnson and Wamsley (1970)) that if G and H are finite p-groups then r(G x H) = r(G) + r(H) + d(G)d(H) and hence G, H∈Gp implies Gx H∈Gp, also it is shown in Wamsley (1972) that if G is any finite pgroup then there exists an H∈Gp such that G x H belongs to Gp. Let G1 = G and Gk = Gk-1 x G then we show in this note that if G is any finite p-group, there exists an integer n(G), such that Gk∈Gp for alal k ≧ n(G).

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