scholarly journals On Normal Subgroups of Generalized Hecke Groups

2016 ◽  
Vol 24 (2) ◽  
pp. 169-184
Author(s):  
Bilal Demir ◽  
Özden Koruoğlu ◽  
Recep Sahin
Keyword(s):  
Commutator Subgroup ◽  
Normal Subgroups ◽  
Hecke Groups

Abstract We consider the generalized Hecke groups Hp,q generated by X(z) = -(z -λp)-1, Y (z) = -(z +λq)-1 with and where 2 ≤ p ≤ q < ∞, p+q > 4. In this work we study the structure of genus 0 normal subgroups of generalized Hecke groups. We construct an interesting genus 0 subgroup called even subgroup, denoted by . We state the relation between commutator subgroup H′p,q of Hp,q defined in [1] and the even subgroup. Then we extend this result to extended generalized Hecke groups H̅p,q.

Normal Subgroups of Hecke Groups H(λ)

2009 ◽  
Vol 13 (2) ◽  
pp. 219-230 ◽  
Author(s):  
Özden Koruoğlu ◽  
Recep Sahin ◽  
Sebahattin İkikardes ◽  
Ismail Naci Cangül
Keyword(s):  
Normal Subgroups ◽  
Hecke Groups

Groups with few conjugacy classes of non-normal subgroups

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Maria Falco ◽  
Francesco Giovanni ◽  
Carmela Musella
Keyword(s):  
Abelian Subgroup ◽  
Soluble Group ◽  
Commutator Subgroup ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
Abelian Subgroups ◽  
Locally Soluble Group

AbstractThe structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

scholarly journals Some notes on Lie ideals in division rings

2018 ◽  
Vol 17 (03) ◽  
pp. 1850049
Author(s):  
M. Aaghabali ◽  
M. Ariannejad ◽  
A. Madadi
Keyword(s):  
Division Ring ◽  
Commutator Subgroup ◽  
Product Formula ◽  
Main Concern ◽  
Normal Subgroups ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Finitely Generated ◽  
Finite Dimensional ◽  
Lie Product Formula

A Lie ideal of a division ring [Formula: see text] is an additive subgroup [Formula: see text] of [Formula: see text] such that the Lie product [Formula: see text] of any two elements [Formula: see text] is in [Formula: see text] or [Formula: see text]. The main concern of this paper is to present some properties of Lie ideals of [Formula: see text] which may be interpreted as being dual to known properties of normal subgroups of [Formula: see text]. In particular, we prove that if [Formula: see text] is a finite-dimensional division algebra with center [Formula: see text] and [Formula: see text], then any finitely generated [Formula: see text]-module Lie ideal of [Formula: see text] is central. We also show that the additive commutator subgroup [Formula: see text] of [Formula: see text] is not a finitely generated [Formula: see text]-module. Some other results about maximal additive subgroups of [Formula: see text] and [Formula: see text] are also presented.

scholarly journals Generation of relative commutator subgroups in Chevalley groups

2015 ◽  
Vol 59 (2) ◽  
pp. 393-410 ◽  
Author(s):  
R. Hazrat ◽  
N. Vavilov ◽  
Z. Zhang
Keyword(s):  
Normal Subgroup ◽  
Chevalley Group ◽  
Commutator Subgroup ◽  
Normal Subgroups ◽  
Elementary Subgroup ◽  
Elementary Group ◽  
Commutator Width ◽  
Relative Commutator ◽  
Elementary Generator ◽  
Elementary Chevalley Group

AbstractLet Φ be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Φ,R,I),E(Φ,R,J)] both as normal subgroups of the elementary Chevalley group E(Φ,R), and as groups. Namely, let xα(ξ), α ∈ Φ ξ ∈ R, be an elementary generator of E(Φ,R). As a normal subgroup of the absolute elementary group E(Φ,R), the relative elementary subgroup is generated by xα(ξ), α ∈ Φ, ξ ∈ I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Φ,R,I) is generated by zα(ξ,η), where α ∈ Φ, ξ ∈ I, η ∈ R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(Φ,R) the relative commutator subgroup [E(Φ,R,I),E(Φ,R,J)] is generated by the following three types of generators: (i) [xα(ξ),zα(ζ,η)], (ii) [xα(ξ),x_α(ζ)] and (iii) xα(ξζ), where α ∈ Φ, ξ ∈ I, ζ ∈ J, η ∈ R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) zα(ξζ,η). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.

EFFICIENT QUANTUM ALGORITHMS FOR SOME INSTANCES OF THE NON-ABELIAN HIDDEN SUBGROUP PROBLEM

2003 ◽  
Vol 14 (05) ◽  
pp. 723-739 ◽  
Author(s):  
GÁBOR IVANYOS ◽  
FRÉDÉRIC MAGNIEZ ◽  
MIKLOS SANTHA
Keyword(s):  
Commutator Subgroup ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Solvable Groups ◽  
Normal Subgroups ◽  
Hidden Subgroup Problem ◽  
Special Cases ◽  
Subgroup Problem ◽  
Small Index ◽  
Hidden Subgroups

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.

Upper Bounds for the Level of Normal Subgroups of Hecke Groups

2011 ◽  
Author(s):  
Musa Demirci ◽  
Aysun Yurttas ◽  
I. Naci Cangul ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
...  
Keyword(s):  
Upper Bounds ◽  
Normal Subgroups ◽  
Hecke Groups

scholarly journals NORMAL SUBGROUPS, CENTER AND INNER AUTOMORPHISMS OF COMPACT QUANTUM GROUPS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350071 ◽  
Author(s):  
ISSAN PATRI
Keyword(s):  
Quantum Groups ◽  
Commutator Subgroup ◽  
Classical Case ◽  
Normal Subgroups ◽  
Compact Quantum Groups ◽  
Inner Automorphisms

We introduce a class of automorphisms of compact quantum groups (CQGs) which may be thought of as inner automorphisms and explore the behavior of normal subgroups of CQGs under these automorphisms. We also define the notion of center of a CQG and compute the center for several examples. We briefly touch upon the commutator subgroup of a CQG and discuss how its relation with the center can be different from the classical case.

scholarly journals Normal subgroups of invertibles and of unitaries in a C*-algebra

2019 ◽  
Vol 2019 (756) ◽  
pp. 285-319
Author(s):  
Leonel Robert
Keyword(s):  
Simple Group ◽  
Commutator Subgroup ◽  
Normal Subgroups ◽  
Connected Component ◽  
C Algebra ◽  
Inner Automorphisms ◽  
Extra Assumption ◽  
Group Modulo

AbstractWe investigate the normal subgroups of the groups of invertibles and unitaries in the connected component of the identity of a {\mathrm{C}^{*}}-algebra. By relating normal subgroups to closed two-sided ideals we obtain a “sandwich condition” describing all the closed normal subgroups both in the invertible and in the unitary case. We use this to prove a conjecture by Elliott and Rørdam: in a simple \mathrm{C}^{*}-algebra, the group of approximately inner automorphisms induced by unitaries in the connected component of the identity is topologically simple. Turning to non-closed subgroups, we show, among other things, that in a simple unital \mathrm{C}^{*}-algebra the commutator subgroup of the group of invertibles in the connected component of the identity is a simple group modulo its center. A similar result holds for unitaries under a mild extra assumption.

scholarly journals On a set of normal subgroups

1962 ◽  
Vol 5 (3) ◽  
pp. 137-146 ◽  
Author(s):  
I. D. Macdonald
Keyword(s):  
Commutator Subgroup ◽  
Normal Subgroups ◽  
Cyclic Subgroups ◽  
Identical Relation

The commutator [a, b] of two elements a and b in a group G satisfies the identityab = ba[a, b].The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation(A) [g, x]Y = [g, y]−1[g, xy]holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that{g}G = {g, K(g)}.

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