On some polycyclic groups with small Hirsch length II

2019 ◽  
Vol 18 (09) ◽  
pp. 1950169
Author(s):  
Heguo Liu ◽  
Fang Zhou ◽  
Tao Xu
Keyword(s):  
Abelian Subgroup ◽  
Polycyclic Group ◽  
Normal Abelian Subgroup ◽  
Number Of Generators ◽  
Finite Quotient ◽  
Polycyclic Groups

A polycyclic group [Formula: see text] is called a [Formula: see text]-group ([Formula: see text]-group) if every normal abelian subgroup (abelian subgroup) of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, we describe the structures of [Formula: see text]-groups and [Formula: see text]-groups, and bound the number of generators of [Formula: see text]-groups and the derived lengths of [Formula: see text]-groups, which is a continuation of [H. G. Liu, F. Zhou and T. Xu, On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710].

On some polycyclic groups with small Hirsch length

2017 ◽  
Vol 16 (12) ◽  
pp. 1750237
Author(s):  
Heguo Liu ◽  
Fang Zhou ◽  
Tao Xu
Keyword(s):  
Abelian Subgroup ◽  
Polycyclic Group ◽  
Normal Abelian Subgroup ◽  
Finite Quotient ◽  
Polycyclic Groups

A polycyclic group [Formula: see text] is called an [Formula: see text]-group if every normal abelian subgroup of any finite quotient of [Formula: see text] is generated by [Formula: see text], or fewer, elements and [Formula: see text] is the least integer with this property. In this paper, the structure of [Formula: see text]-groups and [Formula: see text]-groups is determined.

Bounding the number of generators of a polycyclic group

1981 ◽  
Vol 37 (1) ◽  
pp. 7-17 ◽  
Author(s):  
P. A. Linnell ◽  
D. Warhurst
Keyword(s):  
Polycyclic Group ◽  
Number Of Generators

A Lower Estimate for Central Probabilities on Polycyclic Groups

1992 ◽  
Vol 44 (5) ◽  
pp. 897-910 ◽  
Author(s):  
G. Alexopoulos
Keyword(s):  
Probability Measure ◽  
Lie Group ◽  
Exponential Growth ◽  
Lower Estimate ◽  
Large Time ◽  
Polycyclic Group ◽  
Central Value ◽  
Convolution Power ◽  
Polycyclic Groups

AbstractWe give a lower estimate for the central value μ*n(e) of the nth convolution power μ*···*μ of a symmetric probability measure μ on a polycyclic group G of exponential growth whose support is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (∂/∂t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth.

On the number of minimal nonabelian subgroups in a nonabelian finite p-group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750054
Author(s):  
Yakov Berkovich
Keyword(s):  
Abelian Subgroup ◽  
Normal Abelian Subgroup

The number of minimal nonabelian subgroups in a nonabelian finite [Formula: see text]-group containing a maximal normal abelian subgroup of given index is estimated. Also the [Formula: see text]-group for which the above estimate is attained, are described in great detail.

Groups with Representations of Bounded Degree

1964 ◽  
Vol 16 ◽  
pp. 299-309 ◽  
Author(s):  
I. M. Isaacs ◽  
D. S. Passman
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Abelian Subgroup ◽  
Discrete Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Bounded Degree ◽  
Normal Abelian Subgroup

Let G be a discrete group with group algebra C[G] over the complex numbers C. In (5) Kaplansky essentially proves that if G has a normal abelian subgroup of finite index n, then all irreducible representations of C[G] have degree ≤n. Our main theorem is a converse of Kaplansky's result. In fact we show that if all irreducible representations of C[G] have degree ≤n, then G has an abelian subgroup of index not greater than some function of n. (The degree of a representation of C[G] for arbitrary G is defined precisely in § 3.)

scholarly journals On gauging finite subgroups

2020 ◽  
Vol 8 (1) ◽  
Author(s):  
Yuji Tachikawa
Keyword(s):  
Symmetry Group ◽  
Direct Product ◽  
Abelian Subgroup ◽  
Condensed Matter ◽  
Spacetime Dimension ◽  
Normal Abelian Subgroup ◽  
Condensed Matter Theory ◽  
Matter Theory ◽  
Finite Subgroups ◽  
Symmetric Theory

We study in general spacetime dimension the symmetry of the theory obtained by gauging a non-anomalous finite normal Abelian subgroup AA of a \GammaΓ-symmetric theory. Depending on how anomalous \GammaΓ is, we find that the symmetry of the gauged theory can be i) a direct product of G=\Gamma/AG=Γ/A and a higher-form symmetry \hat AÂ with a mixed anomaly, where \hat AÂ is the Pontryagin dual of AA; ii) an extension of the ordinary symmetry group GG by the higher-form symmetry \hat AÂ; iii) or even more esoteric types of symmetries which are no longer groups. We also discuss the relations to the effect called the H^3(G,\hat A)H3(G,Â) symmetry localization obstruction in the condensed-matter theory and to some of the constructions in the works of Kapustin-Thorngren and Wang-Wen-Witten.

Free subgroups in maximal subgroups of SLn(D)

2020 ◽  
pp. 2150071
Author(s):  
R. Fallah-Moghaddam
Keyword(s):  
Division Ring ◽  
Finite Simple Group ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Central Division ◽  
Normal Abelian Subgroup ◽  
Finite Dimensional ◽  
Free Subgroups

Given a non-commutative finite-dimensional [Formula: see text]-central division ring [Formula: see text], [Formula: see text] a subnormal subgroup of [Formula: see text] and [Formula: see text] a non-abelian maximal subgroup of [Formula: see text], then either [Formula: see text] contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a subfield of [Formula: see text], [Formula: see text] is Galois and [Formula: see text], also [Formula: see text] is a finite simple group with [Formula: see text].

Maximal Abelian Subgroups of the Symmetric Groups

1971 ◽  
Vol 23 (3) ◽  
pp. 426-438 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Symmetric Group ◽  
General Problem ◽  
Abelian Subgroup ◽  
Symmetric Groups ◽  
Specific Class ◽  
Maximal Abelian Subgroup ◽  
Number Of Generators ◽  
Abelian Subgroups ◽  
Almost All

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

On polycyclic groups with isomorphic finite quotients

1978 ◽  
Vol 84 (2) ◽  
pp. 235-246 ◽  
Author(s):  
Fritz Grunewald ◽  
Daniel Segal
Keyword(s):  
Finite Groups ◽  
Polycyclic Group ◽  
Isomorphism Classes ◽  
Finite Quotients ◽  
Polycyclic Groups ◽  
Special Case

Following P. F. Pickel (5) we write (G) for the set of isomorphism classes of finite quotients of a group G. One of the outstanding problems in the theory of polycyclic groups is to determine whether there can be infinitely many non-isomorphic polycyclic groups G with a given (G). We solve a special case of this problem with our first main result:Theorem 1. Let G be an abelian-by-cyclic polycyclic group. Then the polycyclic-by-finite groups H withlie in only finitely many isomorphism classes.

scholarly journals Abelian p-subgroups of the general linear group

1970 ◽  
Vol 11 (3) ◽  
pp. 257-259 ◽  
Author(s):  
J. T. Goozeff
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
Finite Fields ◽  
Abelian Subgroup ◽  
General Linear Group ◽  
General Linear ◽  
Normal Abelian Subgroup ◽  
Abelian Subgroups

A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.

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