scholarly journals Local nearrings on finite non-abelian $2$-generated $p$-groups

2020 ◽  
Vol 12 (1) ◽  
pp. 199-207
Author(s):  
I.Yu. Raievska ◽  
M.Yu. Raievska
Keyword(s):  
Commutator Subgroup ◽  
Additive Group ◽  
Nilpotency Class ◽  
Invertible Elements ◽  
Cyclic Commutator

It is proved that for ${p>2}$ every finite non-metacyclic $2$-generated p-group of nilpotency class $2$ with cyclic commutator subgroup is the additive group of a local nearring and in particular of a nearring with identity. It is also shown that the subgroup of all non-invertible elements of this nearring is of index $p$ in its additive group.

scholarly journals Subgroups of finite index in an additive group of a ring

2001 ◽  
Vol 27 (2) ◽  
pp. 83-89
Author(s):  
Doostali Mojdeh ◽  
S. Hassan Hashemi
Keyword(s):  
Finite Index ◽  
Additive Group ◽  
Infinite Field ◽  
Division Rings ◽  
Invertible Elements

IfKis an infinite field andG⫅Kis a subgroup of finite index in an additive group, thenK∗=G∗G∗−1whereG∗denotes the set of all invertible elements inGandG∗−1denotes all inverses of elements ofG∗. Similar results hold for various fields, division rings and rings.

scholarly journals Computing the Nonabelian Tensor Square of Metacyclic p–Groups of Nilpotency Class Two

2013 ◽  
Vol 61 (1) ◽  
Author(s):  
A. M. Basri ◽  
N. H. Sarmin ◽  
N. M. Mohd Ali ◽  
J. R. Beuerle
Keyword(s):  
Commutator Subgroup ◽  
Group Structure ◽  
Nilpotency Class ◽  
Conjugacy Classes ◽  
Schur Multiplier ◽  
Group Order ◽  
Nonabelian Tensor ◽  
Current Article ◽  
Tensor Square ◽  
Compute Structure

In this paper, we develop appropriate programme using Groups, Algorithms and Programming (GAP) software enables performing different computations on various characteristics of all finite nonabelian metacyclic p–groups, p is prime, of nilpotency class 2. Such programme enables to compute structure of the group, order of the group, structure of the center, the number of conjugacy classes, structure of commutator subgroup, abelianization, Whitehead’s universal quadratic functor and other characteristics. In addition, structures of some other groups such as the nonabelian tensor square and various homological functors including Schur multiplier and exterior square can be computed using this programme. Furthermore, by computing the epicenter order or the exterior center order the capability can be determined. In our current article, we only compute the nonabelian tensor square of certain order groups, as an example, and give GAP codes for computing other characteristics and some subgroups.

A particular class of supersoluble groups

1994 ◽  
Vol 57 (3) ◽  
pp. 357-364 ◽  
Author(s):  
W. Dirscherl ◽  
H. Heineken
Keyword(s):  
Finite Groups ◽  
Quotient Group ◽  
Commutator Subgroup ◽  
Supersoluble Groups ◽  
Cyclic Commutator

AbstractWe consider (finite) groups in which every two-generator subgroup has cyclic commutator subgroup. Among other things, these groups are metabelian modulo their hypercentres, and in the corresponding quotient group all subgroups of the commutator subgroup are normal.

On finitep-groups with cyclic commutator subgroup

1982 ◽  
Vol 39 (4) ◽  
pp. 295-298 ◽  
Author(s):  
Ying Cheng
Keyword(s):  
Commutator Subgroup ◽  
Cyclic Commutator

Finitep-groups with a cyclic commutator subgroup

1995 ◽  
Vol 34 (2) ◽  
pp. 125-129 ◽  
Author(s):  
A. A. Finogenov
Keyword(s):  
Commutator Subgroup ◽  
Cyclic Commutator

scholarly journals On Finite Quasi-Core-p p-Groups

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1147
Author(s):  
Jiao Wang ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Commutator Subgroup ◽  
Nilpotency Class ◽  
Frattini Subgroup ◽  
Normal Core ◽  
A Finite Group

Given a positive integer n, a finite group G is called quasi-core-n if ⟨ x ⟩ / ⟨ x ⟩ G has order at most n for any element x in G, where ⟨ x ⟩ G is the normal core of ⟨ x ⟩ in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.

scholarly journals Automorphism groups with cyclic commutator subgroup and Hamilton cycles

1998 ◽  
Vol 189 (1-3) ◽  
pp. 69-78 ◽  
Author(s):  
Edward Dobson ◽  
Heather Gavlas ◽  
Joy Morris ◽  
Dave Witte
Keyword(s):  
Commutator Subgroup ◽  
Automorphism Groups ◽  
Hamilton Cycles ◽  
Cyclic Commutator

On p-groups with a cyclic commutator subgroup

1975 ◽  
Vol 20 (2) ◽  
pp. 178-198 ◽  
Author(s):  
R. J. Miech
Keyword(s):  
Commutator Subgroup ◽  
Metabelian Group ◽  
Special Kind ◽  
Complete Classification ◽  
Isomorphism Problem ◽  
Point Of View ◽  
Defining Relations ◽  
Group A ◽  
Cyclic Commutator

This paper contains the complete classification of the finite p-groups G where p is an odd prime, G is generated by two elements, and the commutator subgroup of G is cyclic. These groups are a special kind of two-generator metabelian group, a class that has been studied by Szekeres (1965). He determined the defining relations of such groups but, as he noted, a “residual isomorphism problem” remains. The cyclic commutator groups are simple when considered from this first point of view; they have a short, easily derived set of defining relations. However, the isomorphism problem is a bit complicated for the defining relations contain nine parameters and each of these parameters might and can be an invariant of the group.

Sign in / Sign up

Export Citation Format

Share Document