scholarly journals Metabelian groups: Full-rank presentations, randomness and Diophantine problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Albert Garreta ◽  
Leire Legarreta ◽  
Alexei Miasnikov ◽  
Denis Ovchinnikov
Keyword(s):  
Normal Subgroup ◽  
Full Rank ◽  
Direct Decomposition ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Diophantine Problem ◽  
Diophantine Problems ◽  
Finitely Presented ◽  
Finite Presentations

AbstractWe study metabelian groups 𝐺 given by full rank finite presentations \langle A\mid R\rangle_{\mathcal{M}} in the variety ℳ of metabelian groups. We prove that 𝐺 is a product of a free metabelian subgroup of rank \max\{0,\lvert A\rvert-\lvert R\rvert\} and a virtually abelian normal subgroup, and that if \lvert R\rvert\leq\lvert A\rvert-2, then the Diophantine problem of 𝐺 is undecidable, while it is decidable if \lvert R\rvert\geq\lvert A\rvert. We further prove that if \lvert R\rvert\leq\lvert A\rvert-1, then, in any direct decomposition of 𝐺, all factors, except one, are virtually abelian. Since finite presentations have full rank asymptotically almost surely, metabelian groups finitely presented in the variety of metabelian groups satisfy all the aforementioned properties asymptotically almost surely.

Degrees of faithful irreducible representations of metabelian groups

2021 ◽  
pp. 2250181
Author(s):  
Rahul Dattatraya Kitture ◽  
Soham Swadhin Pradhan
Keyword(s):  
Normal Subgroup ◽  
Finite Groups ◽  
Positive Characteristic ◽  
Number Fields ◽  
Irreducible Representations ◽  
Modular Representations ◽  
Metacyclic Group ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Field Of Positive Characteristic

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

TTHE CONJUGACY CLASSES OF METABELIAN GROUPS OF ORDER AT MOST 24

2015 ◽  
Vol 77 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ibrahim Gambo ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Equivalence Relation ◽  
Metabelian Group ◽  
Equivalence Classes ◽  
Factor Group ◽  
Conjugacy Classes ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup

In this paper, G denotes a non-abelian metabelian group and denotes conjugacy class of the element x in G. Conjugacy class is an equivalence relation and it partitions the group into disjoint equivalence classes or sets. Meanwhile, a group is called metabelian if it has an abelian normal subgroup in which the factor group is also abelian. It has been proven by an earlier researcher that there are 25 non-abelian metabelian groups of order less than 24 which are considered in this paper. In this study, the number of conjugacy classes of non-abelian metabelian groups of order less than 24 is computed.

scholarly journals On Some Graphs of Finite Metabelian Groups of Order Less Than 24

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 237-247
Author(s):  
Ibrahim Gambo ◽  
Nor Haniza Sarmin ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Normal Subgroup ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Metabelian Group ◽  
Factor Group ◽  
Conjugacy Classes ◽  
Metabelian Groups ◽  
Abelian Normal Subgroup ◽  
Class Graph

In this work, a non-abelian metabelian group is represented by G while represents conjugacy class graph. Conjugacy class graph of a group is that graph associated with the conjugacy classes of the group. Its vertices are the non-central conjugacy classes of the group, and two distinct vertices are joined by an edge if their cardinalities are not coprime. A group is referred to as metabelian if there exits an abelian normal subgroup in which the factor group is also abelian. It has been proven earlier that 25 non-abelian metabelian groups which have order less than 24, which are considered in this work, exist. In this article, the conjugacy class graphs of non-abelian metabelian groups of order less than 24 are determined as well as examples of some finite groups associated to other graphs are given.

scholarly journals Electric-magnetic duality in twisted quantum double model of topological orders

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Yuting Hu ◽  
Yidun Wan
Keyword(s):  
Normal Subgroup ◽  
Gauge Group ◽  
The Other ◽  
Quantum Double ◽  
Duality Transformation ◽  
Abelian Normal Subgroup

Abstract We derive a partial electric-magnetic (PEM) duality transformation of the twisted quantum double (TQD) model TQD(G, α) — discrete Dijkgraaf-Witten model — with a finite gauge group G, which has an Abelian normal subgroup N , and a three-cocycle α ∈ H3(G, U(1)). Any equivalence between two TQD models, say, TQD(G, α) and TQD(G′, α′), can be realized as a PEM duality transformation, which exchanges the N-charges and N-fluxes only. Via the PEM duality, we construct an explicit isomorphism between the corresponding TQD algebras Dα(G) and Dα′(G′) and derive the map between the anyons of one model and those of the other.

scholarly journals Metabelian groups with the same finite quotients

1974 ◽  
Vol 11 (1) ◽  
pp. 115-120 ◽  
Author(s):  
P.F. Pickel
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Isomorphism Classes ◽  
Split Extensions ◽  
Finite Quotients ◽  
Finitely Presented

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. Two groups G and H are said to have the same finite quotients if F(G) = F(H). We construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring. These groups also give an example of infinitely many nonisomorphic split extensions of a fixed finitely presented metabelian group by a fixed finite abelian group, all having the same finite quotients.

scholarly journals Fitting quotients of finitely presented abelian-by-nilpotent groups

2014 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
J. R. J. Groves ◽  
Ralph Strebel
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Presented

Abstract.We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

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