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.

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 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.

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.

A Constructive Brauer-Witt Theorem for Certain Solvable Groups

1996 ◽  
Vol 48 (6) ◽  
pp. 1196-1209 ◽  
Author(s):  
Allen Herman
Keyword(s):  
Finite Groups ◽  
Solvable Groups ◽  
Number Fields ◽  
Prime Integer ◽  
Algebraic Number Fields ◽  
Metabelian Groups ◽  
Elementary Subgroup ◽  
Simple Component ◽  
Clifford Theory ◽  
Simple Algebras

AbstractDivision algebras occurring in simple components of group algebras of finite groups over algebraic number fields are studied. First, well-known restrictions are presented for the structure of a group that arises once no further Clifford Theory reductions are possible. For groups with these properties, a character-theoretic condition is given that forces the p-part of the division algebra part of this simple component to be generated by a predetermined p-quasi-elementary subgroup of the group, for any prime integer p. This is effectively a constructive Brauer-Witt Theorem for groups satisfying this condition. It is then shown that it is possible to constructively compute the Schur index of a simple component of the group algebra of a finite nilpotent-by-abelian group using the above reduction and an algorithm for computing Schur indices of simple algebras generated by finite metabelian groups.

scholarly journals Oort groups and lifting problems

2008 ◽  
Vol 144 (4) ◽  
pp. 849-866 ◽  
Author(s):  
T. Chinburg ◽  
R. Guralnick ◽  
D. Harbater
Keyword(s):  
Finite Groups ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Cyclic Groups ◽  
Characteristic Two ◽  
Field Of Positive Characteristic

AbstractLet k be an algebraically closed field of positive characteristic p. We consider which finite groups G have the property that every faithful action of G on a connected smooth projective curve over k lifts to characteristic zero. Oort conjectured that cyclic groups have this property. We show that if a cyclic-by-p group G has this property, then G must be either cyclic or dihedral, with the exception of A4 in characteristic two. This proves one direction of a strong form of the Oort conjecture.

scholarly journals On minimal faithful permutation representations of finite groups

2000 ◽  
Vol 62 (2) ◽  
pp. 311-317 ◽  
Author(s):  
L. G. Kovács ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Symmetric Group ◽  
Finite Groups ◽  
Abelian Normal Subgroup ◽  
Minimal Counterexample ◽  
Representations Of Finite Groups ◽  
A Finite Group

The minimal faithful permutation degree μ(G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group Sn. Let N be a normal subgroup of a finite group G. We prove that μ(G/N) ≤ μ(G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate the structure of a (hypothetical) minimal counterexample to this conjecture.

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