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.

On the number of generators of a finite group

1989 ◽  
Vol 53 (6) ◽  
pp. 521-523 ◽  
Author(s):  
Robert M. Guralnick
Keyword(s):  
Finite Group ◽  
Number Of Generators ◽  
A Finite Group

scholarly journals Finite p–nilpotent groups with some subgroups c–supplemented

2005 ◽  
Vol 78 (3) ◽  
pp. 429-439 ◽  
Author(s):  
Xiuyun Guo ◽  
K. P. Shum
Keyword(s):  
Finite Group ◽  
Prime Factor ◽  
Nilpotent Groups ◽  
A Finite Group

AbstractA subgroup H of a finite group G is said to be c–supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is contained in coreG (H). In this paper some results for finite p–nilpotent groups are given based on some subgroups of Pc–supplemented in G, where p is a prime factor of the order of G and P is a Sylow p–subgroup of G. We also give some applications of these results.

scholarly journals Extensions of finite nilpotent groups

1970 ◽  
Vol 2 (2) ◽  
pp. 267-274
Author(s):  
John Poland
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Theoretic Property ◽  
Joint Paper ◽  
The Core ◽  
A Finite Group

If G is a finite group and P is a group-theoretic property, G will be called P-max-core if for every maximal subgroup M of G, M/MG has property P where MG = ∩ is the core of M in G. In a joint paper with John D. Dixon and A.H. Rhemtulla, we showed that if p is an odd prime and G is (p-nilpotent)-max-core, then G is p-solvable, and then using the techniques of the theory of solvable groups, we characterized nilpotent-max-core groups as finite nilpotent-by-nilpotent groups. The proof of the first result used John G. Thompson's p-nilpotency criterion and hence required p > 2. In this paper I show that supersolvable-max-core groups (and hence (2-nilpotent)-max-core groups) need not be 2-solvable (that is, solvable). Also I generalize the second result, among others, and characterize (p-nilpotent)-max-core groups (for p an odd prime) as finite nilpotent-by-(p-nilpotent) groups.

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

A Note on Special Local 2-Nilpotent Groups and the Solvability of Finite Groups

2018 ◽  
Vol 25 (04) ◽  
pp. 541-546
Author(s):  
Jiangtao Shi ◽  
Klavdija Kutnar ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Quaternion Group ◽  
A Finite Group

A finite group G is called a special local 2-nilpotent group if G is not 2-nilpotent, the Sylow 2-subgroup P of G has a section isomorphic to the quaternion group of order 8, [Formula: see text] and NG(P) is 2-nilpotent. In this paper, it is shown that SL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if [Formula: see text], and that GL2(q), [Formula: see text], is a special local 2-nilpotent group if and only if q is odd. Moreover, the solvability of finite groups is also investigated by giving two generalizations of a result from [A note on p-nilpotence and solvability of finite groups, J. Algebra 321 (2009) 1555–1560].

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

On the 𝔉-hypercenter and the intersection of 𝔉-maximal subgroups of a finite group

2018 ◽  
Vol 21 (3) ◽  
pp. 463-473
Author(s):  
Viachaslau I. Murashka
Keyword(s):  
Finite Group ◽  
Nilpotent Groups ◽  
Chief Factor ◽  
Maximal Subgroups ◽  
Inner Automorphism ◽  
Formula Group ◽  
Saturated Formations ◽  
A Finite Group

Abstract Let {\mathfrak{X}} be a class of groups. A subgroup U of a group G is called {\mathfrak{X}} -maximal in G provided that (a) {U\in\mathfrak{X}} , and (b) if {U\leq V\leq G} and {V\in\mathfrak{X}} , then {U=V} . A chief factor {H/K} of G is called {\mathfrak{X}} -eccentric in G provided {(H/K)\rtimes G/C_{G}(H/K)\not\in\mathfrak{X}} . A group G is called a quasi- {\mathfrak{X}} -group if for every {\mathfrak{X}} -eccentric chief factor {H/K} and every {x\in G} , x induces an inner automorphism on {H/K} . We use {\mathfrak{X}^{*}} to denote the class of all quasi- {\mathfrak{X}} -groups. In this paper we describe all hereditary saturated formations {\mathfrak{F}} containing all nilpotent groups such that the {\mathfrak{F}^{*}} -hypercenter of G coincides with the intersection of all {\mathfrak{F}^{*}} -maximal subgroups of G for every group G.

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