Commutator Subgroup Research Articles

Left multiplication operators on the Riordan group

In this paper, we introduce the left multiplication operators on the Riordan group, and carry out a systematic study on them. By using these operators, we obtain a family H of infinite Riordan subgroups. This family includes not only some well-known subgroups, but also the subgroup families introduced recently in literature. Fundamental properties of the left multiplication operators and the subgroup family H are presented. The Riordan involutions and pseudo-involutions, the stabilizer subgroups in family H, and the n-th commutator subgroup of any subgroup in family H are studied. Some further applications of the left multiplication operators are also provided. In particular, we study a class of left multiplication operators which are closely related to the diagonal translation operator and the [m]-complementary operator, and enrich the theory of complementary Riordan arrays. Moreover, we introduce and study the higher-level lifts of Riordan arrays as well as the vertical and horizontal 1/m Riordan arrays. As a result, it can be found that the left multiplication operators provide a unified approach to numerous subgroups, operators and transformations of the Riordan group.

Commutators and commutator subgroups of the Riordan group

We calculate the derived series of the Riordan group. To do this, we study a nested sequence of its subgroups, herein denoted by Gk. By means of this sequence, we first obtain the n-th commutator subgroup of the Associated subgroup. This fact allows us to get some related results about certain groups of formal power series and to complete the proof of our main goal, Theorem 1 in this paper.

Virtual braid groups, virtual twin groups and crystallographic groups

Let n≥2. Let VBn (resp. VPn) be the virtual braid group (resp. the pure virtual braid group), and let VTn (resp. PVTn) be the virtual twin group (resp. the pure virtual twin group). Let Π be one of the following quotients: VBn/Γ2(VPn) or VTn/Γ2(PVTn) where Γ2(H) is the commutator subgroup of H. In this paper, we show that Π is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in Π. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in Π. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) modulo the respective commutator subgroup in each case.

On the Structure of the Mislin Genus of a Pullback

The notion of genus for finitely generated nilpotent groups was introduced by Mislin. Two finitely generated nilpotent groups Q and R belong to the same genus set G(Q) if and only if the two groups are nonisomorphic, but for each prime p, their p-localizations Qp and Rp are isomorphic. Mislin and Hilton introduced the structure of a finite abelian group on the genus if the group Q has a finite commutator subgroup. In this study, we consider the class of finitely generated infinite nilpotent groups with a finite commutator subgroup. We construct a pullback Ht from the l-equivalences Hi→H and Hj→H, t≡(i+j)mods, where s=|G(H)|, and compare its genus to that of H. Furthermore, we consider a pullback L of a direct product G×K of groups in this class. Here, we prove results on the group L and prove that its genus is nontrivial.

Study on Poisson Algebra and Automorphism of a Special Class of Solvable Lie Algebras

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀x,y∈g,[x,y]=−[y,x], that is, the operation [,] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by Aut(g)=G3G1G2G3G4G7G8G5, or Aut(g)=G3G1G2G3G4G7G8G5G6, or Aut(g)=G3G1G2G3G4G7G8G5G3, where Gi is a commutative subgroup of Aut(g). We give some subgroups of g’s automorphism group and systematically studied the properties of these subgroups.

On compact groups with Engel-like conditions

Let G be a compact Hausdorff group. We show that if for every element there exists a positive integer such that xq is Engel, then G is locally virtually nilpotent. Furthermore, we show that if G is a finitely generated compact Hausdorff group in which any commutator in G is Engel, then the commutator subgroup is locally nilpotent.

The commutator subgroup of the braid group is generated by two elements

For n n at least 7 and n n equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on n n strands. These generating sets are of the smallest possible cardinality. For n n equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid group on 4 strands cannot be generated by fewer than three elements.

On the second homotopy group of the classifying space for commutativity in Lie groups

In this note we show that the second homotopy group of B(2, G), the classifying space for commutativity for a compact Lie group G, contains a direct summand isomorphic to $$\pi _1(G)\oplus \pi _1([G,G])$$ , where [G, G] is the commutator subgroup of G. It follows from a similar statement for E(2, G), the homotopy fiber of the canonical inclusion $$B(2,G)\hookrightarrow BG$$ . As a consequence of our main result we obtain that if E(2, G) is 2-connected, then [G, G] is simply-connected. This last result completes how the higher connectivity of E(2, G) resembles the higher connectivity of [G, G] for a compact Lie group G.

The Ma–Qiu index and the Nakanishi index for a fibered knot are equal, and ω-solvability

For a knot [Formula: see text] in [Formula: see text], let [Formula: see text] be the knot group of [Formula: see text], [Formula: see text] the Ma–Qiu (MQ) index, which is the minimal number of normal generators of the commutator subgroup of [Formula: see text], and [Formula: see text] the Nakanishi index of [Formula: see text], which is the minimal number of generators of the Alexander module of [Formula: see text]. We generalize the notions for a pair of a group [Formula: see text] and its normal subgroup [Formula: see text], and we denote them by [Formula: see text] and [Formula: see text] respectively. Then it is easy to see [Formula: see text] in general. We also introduce a notion “[Formula: see text]-solvability” for a group that the intersection of all higher commutator subgroups is trivial. Our main theorem is that if [Formula: see text] is [Formula: see text]-solvable, then we have [Formula: see text]. As corollaries, for a fibered knot [Formula: see text], we have [Formula: see text], and we could determine the MQ indices of prime knots up to nine crossings completely.

Subsymmetric exchanged braids and the Burau matrix

We develop a method based on the Burau matrix to detect conditions on the linking numbers of braid strands. Our main application is to iterated exchanged braids. Unless the braid permutation fixes both braid edge strands, we establish under some fairly generic conditions on the linking numbers a ‘subsymmetry’ property; in particular at most two such braids can be mutually conjugate. As an addition, we prove that the Burau kernel is contained in the commutator subgroup of the pure braid group. We discuss also some properties of the Burau image.

On various types of nilpotency of the structure monoid and group of a set-theoretic solution of the Yang–Baxter equation

Given a finite bijective non-degenerate set-theoretic solution ( X , r ) of the Yang–Baxter equation we characterize when its structure monoid M ( X , r ) is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions ( X , r ) and show, for example, that this property is equivalent to the solution associated to the structure monoid M ( X , r ) (respectively structure group G ( X , r ) ) being a multipermuation solution and that G ( X , r ) is solvable of derived length not exceeding the multipermutation level of ( X , r ) enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the square-free involutive case. Moreover, we also prove that if X is finite and G = G ( X , r ) is nilpotent, then the torsion part of the group G is finite, it coincides with the commutator subgroup [ G , G ] + of the additive structure of the skew left brace G and G / [ G , G ] + is a trivial left brace.

Simplicity, bounded normal generation, and automatic continuity of groups of unitaries

We show that the commutator subgroup of the group of unitaries connected to the identity in a simple unital C*-algebra is simple modulo its center. We then go on to investigate the role of “regularity properties” in the structure of the special unitary group of a C*-algebra. Under mild assumptions, we show that this group has the invariant automatic continuity property and a unique polish group topology. Strengthening our assumptions in the case of simple C*-algebras, we show that the special unitary group modulo its center has bounded normal generation. These results apply to all simple purely infinite C*-algebras and to all simple nuclear C*-algebras in the “classifiable class”. We show with counterexamples how our conclusions may in general fail if no regularity conditions are imposed on the C*-algebra.

Residual torsion-free nilpotence, bi-orderability, and two-bridge links

AbstractResidual torsion-free nilpotence has proved to be an important property for knot groups with applications to bi-orderability and ribbon concordance. Mayland proposed a strategy to show that a two-bridge knot group has a commutator subgroup which is a union of an ascending chain of para-free groups. This paper proves Mayland’s assertion and expands the result to the subgroups of two-bridge link groups that correspond to the kernels of maps to $\mathbb{Z}$ . We call these kernels the Alexander subgroups of the links. As a result, we show the bi-orderability of a large family of two-bridge link groups. This proof makes use of a modified version of a graph-theoretic construction of Hirasawa and Murasugi in order to understand the structure of the Alexander subgroup for a two-bridge link group.

On the classification of sub-Riemannian structures on a 5D two-step nilpotent Lie group

We classify the left-invariant sub-Riemannian structures on the unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D group is the first twostep nilpotent Lie group beyond the three-and five-dimensional Heisenberg groups. Alongside, we also present a classification, up to automorphism, of the subspaces of the associated Lie algebra (together with a complete set of invariants).

Random subcomplexes of finite buildings, and fibering of commutator subgroups of right‐angled Coxeter groups

AbstractThe main theme of this paper is higher virtual algebraic fibering properties of right‐angled Coxeter groups (RACGs), with a special focus on those whose defining flag complex is a finite building. We prove for particular classes of finite buildings that their random induced subcomplexes have a number of strong properties, most prominently that they are highly connected. From this we are able to deduce that the commutator subgroup of a RACG, with defining flag complex a finite building of a certain type, admits an epimorphism to whose kernel has strong topological finiteness properties. We additionally use our techniques to present examples where the kernel is of type but not , and examples where the RACG is hyperbolic and the kernel is finitely generated and non‐hyperbolic. The key tool we use is a generalization of an approach due to Jankiewicz–Norin–Wise involving Bestvina–Brady discrete Morse theory applied to the Davis complex of a RACG, together with some probabilistic arguments.

Strongly Regular Relations Derived from Fundamental Relation

We introduce a new regular relation δ on a given group G and show that δ is a congurence relation on G, with respect to module the commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ, and we conclude that, if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of δ on ρ, results in a strongly regular relation on H such that its quotient is an abelian group.

Compact groups with probabilistically central monothetic subgroups

If K is a closed subgroup of a compact group G, the probability that a randomly chosen pair of elements from K and G commute is denoted by Pr(K, G). Say that a subgroup K ≤ G is ϵ-central in G if Pr(〈g〉, G) ≥ ϵ for any g in K. Here 〈g〉 denotes the monothetic subgroup generated by g ∈ G. Our main result is that if K is ϵ-central in G, then there is an ϵ-bounded number e and a normal subgroup T ≤ G such that both the index [G:T] and the order of the commutator subgroup [Ke, T] are finite and ϵ-bounded. In particular, if G is a compact group for which there is ϵ > 0 such that Pr(〈g〉, G) ≥ ϵ for any g ∈ G, then there is an ϵ-bounded number e and a normal subgroup T such that both the index [G:T] and the order of [Ge, T] are finite and ϵ-bounded.

On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic

Let p be a an odd prime and let G be a finite p-group with cyclic commutator subgroup G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document}. We prove that the exponent and the abelianization of the centralizer of G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document} in G are determined by the group algebra of G over any field of characteristic p. If, additionally, G is 2-generated then almost all the numerical invariants determining G up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G^{\\prime }$\\end{document} is determined. These claims are known to be false for p = 2.

Commutators of relative and unrelative elementary unitary groups

In the present paper, which is an outgrowth of the authors’ joint work with Anthony Bak and Roozbeh Hazrat on the unitary commutator calculus [9, 27, 30, 31], generators are found for the mixed commutator subgroups of relative elementary groups and unrelativized versions of commutator formulas are obtained in the setting of Bak’s unitary groups. It is a direct sequel of the papers [71, 76, 78, 79] and [77, 80], where similar results were obtained forGL(n,R)GL(n,R)and for Chevalley groups over a commutative ring with 1, respectively. Namely, let(A,Λ)(A,\Lambda )be any form ring and letn≥3n\ge 3. Bak’s hyperbolic unitary groupGU(2n,A,Λ)GU(2n,A,\Lambda )is considered. Further, let(I,Γ)(I,\Gamma )be a form ideal of(A,Λ)(A,\Lambda ). One can associate with the ideal(I,Γ)(I,\Gamma )the corresponding true elementary subgroupFU(2n,I,Γ)FU(2n,I,\Gamma )and the relative elementary subgroupEU(2n,I,Γ)EU(2n,I,\Gamma )ofGU(2n,A,Λ)GU(2n,A,\Lambda ). Let(J,Δ)(J,\Delta )be another form ideal of(A,Λ)(A,\Lambda ). In the present paper an unexpected result is proved that the nonobvious type of generators for[EU(2n,I,Γ),EU(2n,J,Δ)]\big [EU(2n,I,\Gamma ),EU(2n,J,\Delta )\big ], as constructed in the authors’ previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugatesZij(ξ,c)=Tji(c)Tij(ξ)Tji(−c)Z_{ij}(\xi ,c)=T_{ji}(c)T_{ij}(\xi )T_{ji}(-c), and the elementary commutatorsYij(a,b)=[Tij(a),Tji(b)]Y_{ij}(a,b)=[T_{ij}(a),T_{ji}(b)], wherea∈(I,Γ)a\in (I,\Gamma ),b∈(J,Δ)b\in (J,\Delta ),c∈(A,Λ)c\in (A,\Lambda ), andξ∈(I,Γ)∘(J,Δ)\xi \in (I,\Gamma )\circ (J,\Delta ). It follows that[FU(2n,I,Γ),FU(2n,J,Δ)]=[EU(2n,I,Γ),EU(2n,J,Δ)]\big [FU(2n,I,\Gamma ),FU(2n,J,\Delta )\big ]=\big [EU(2n,I,\Gamma ),EU(2n,J,\Delta )\big ]. In fact, much more precise generation results are established. In particular, even the elementary commutatorsYij(a,b)Y_{ij}(a,b)should be taken for one long root position and one short root position. Moreover, theYij(a,b)Y_{ij}(a,b)are central moduloEU(2n,(I,Γ)∘(J,Δ))EU(2n,(I,\Gamma )\circ (J,\Delta ))and behave as symbols. This makes it possible to generalize and unify many previous results, including the multiple elementary commutator formula, and dramatically simplify their proofs.

On the Bieri–Neumann–Strebel–Renz invariants of the weak commutativity construction $\mathfrak{X}(G)$

For a finitely generated group G , we calculate the Bieri–Neumann–Strebel–Renz invariant \Sigma^1(\mathfrak{X}(G)) for the weak commutativity construction \mathfrak{X}(G) . Identifying S(\mathfrak{X}(G)) with S(\mathfrak{X}(G) /\allowbreak W(G)) , we show \Sigma^2(\mathfrak{X}(G),\mathbb{Z}) \subseteq \Sigma^2(\mathfrak{X}(G)/ W(G),\mathbb{Z}) and \Sigma^2(\mathfrak{X}(G)) \subseteq \Sigma^2(\mathfrak{X}(G)/ W(G)) , that are equalities when W(G) is finitely generated, and we explicitly calculate \Sigma^2(\mathfrak{X}(G)/ W(G),\mathbb{Z}) and \Sigma^2(\mathfrak{X}(G)/ W(G)) in terms of the \Sigma -invariants of G . We calculate completely the \Sigma -invariants in dimensions 1 and 2 of the group \nu(G) and show that if G is finitely generated group with finitely presented commutator subgroup then the non-abelian tensor square G \otimes G is finitely presented.