Commutator Subgroup Research Articles

A NOTE ON DERIVED LENGTH AND CHARACTER DEGREES

AbstractIsaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$. We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

Finite Groups with $$\mathfrak F$$-Subnormal Subgroups

Let $$G$$ be a finite group, let $$M$$ be a maximal subgroup of $$G$$ , and let $$\mathfrak F$$ be a hereditary formation consisting of solvable groups. The metanilpotency of the $$\mathfrak F$$ -residual $$G^\mathfrak F$$ is established under the assumption that all subgroups maximal in $$M$$ are $$\mathfrak F$$ -subnormal in $$G$$ , and the nilpotency of $$G^\mathfrak F$$ is established in the case where $$\mathfrak F$$ is saturated. Properties of the group $$G$$ are indicated in more detail for the formation of all solvable groups with Abelian Sylow subgroups, for the formation of all supersolvable groups, and for the formation of all groups with nilpotent commutator subgroup.

Local nearrings on finite non-abelian $2$-generated $p$-groups

It is proved that for ${p>2}$ every finite non-metacyclic $2$-generated p-group of nilpotency class $2$ with cyclic commutator subgroup is the additive group of a local nearring and in particular of a nearring with identity. It is also shown that the subgroup of all non-invertible elements of this nearring is of index $p$ in its additive group.

Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band

Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author \cite{SkVC, SkMal, SkAr} and construct minimal generating set for wreath product of finite and infinite cyclic groups and direct product of such groups. We generalize results of Meldrum about commutator subgroup of wreath product \cite{Meld} because we take in consideration as regular wreath product as well as no regular (where active group $\mathcal{A}$ can acts not faithfully). Also commutator of such group and its minimal generating set. Also center of such products was investigated. Also fundamental group of orbits of a Morse function $f:M\to \mathbb{R}$ defined on a Mebius band $M$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$ is investigated by us. The paper describes precise algebraic structure of the group $\pi_1 O(f)$. A minimal set of generators for the group of orbits of functions ${{\pi }_{1}}({{O}_{f}},f)$ arising under the action of diffeomorphisms group stabilizing the function $f$ and stabilizing $\partial M$ is found. The the Morse function $f$ has critical sets with one saddle point. The quotient group of restricted wreath products by its commutator was found. The generic sets of commutator of wreath product were investigated. Minimal generating set for this group and for commutator of group are found. This paper after previous Arxiv versions from 2019 \cite{SkArM, SkArM3} with previous title "Minimal generating set and structure of wreath product of groups with non-faithful action, comutator subgroup of wreath product and the fundamental group of orbit of Morse function $\pi_1 O(f)$" was published \cite{SkRendi}.

An analog of nilpotence arising from supercharacter theory

The goal of this paper is to generalize several group theoretic concepts such as the center and commutator subgroup, central series, and ultimately nilpotence to a supercharacter theoretic setting, and to use these concepts to show that there can be a strong connection between the structure of a group and the structure of its supercharacter theories. We then use these concepts to show that the upper and lower annihilator series of J can be described in terms of certain central series for the algebra group G=1+J defined by S, when S is the algebra group supercharacter theory defined by Diaconis–Isaacs.

Exact β-Function in Abelian and non-Abelian $${\cal N} = 1$$ Supersymmetric Gauge Models and Its Analogy with the QCD β-Function in the C-scheme

For $${\cal N} = 1$$ supersymmetric Yang—Mills theory without matter it is demonstrated that there is a class of renormalization schemes, in which the exact Novikov, Shifman, Vainshtein, and Zakharov (NSVZ) formula for the renormalization group β-function, defined in terms of the renormalized coupling constant, is valid. These schemes are related with each other by finite renormalizations forming a one-parameter commutative subgroup of general renormalization group transformations. The analogy between the exact β-function in $${\cal N} = 1$$ supersymmetric Yang-Mills theory without matter and the β-function of quantum chromodynamics in the C-scheme is discussed.

Higher dimensional generalizations of the Thompson groups

Motivated by the goal of generalizing the Cuntz-Krieger algebras, and making heavy use of the pioneering work of Robertson and Steger, Kumjian and Pask generalized the notion of a directed graph to what they termed a ‘higher rank graph’. The C⁎-algebras constructed from such higher rank graphs have proved to be highly interesting. Higher rank graphs are, in fact, a class of cancellative categories and so the one-vertex higher rank graphs are therefore a class of cancellative monoids — in this paper we call them k-monoids. Finite direct products of free monoids belong to this family but there are many examples not of this form. It is well-known that the classical Thompson groups arise from the free monoids, and it is easy to show that Brin's higher dimensional Thompson groups arise from finite direct products of free monoids. In this paper, we generalize these constructions and show how to construct a family of new groups with simple commutator subgroups from arbitrary aperiodic k-monoids thereby subsuming all the above examples. Although we give simple direct constructions of our groups, a key feature of our paper is that we also show that they arise as topological full groups of the usual étale groupoids associated with k-monoids. In this way, our groups are naturally associated with C⁎-algebras. The question of the mutual isomorphisms amongst this family of groups appears to be a delicate one and is not handled here.

The Derived Subgroups of Sylow 2-Subgroups of the Alternating Group, Commutator Width of Wreath Product of Groups

The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group A 2 k is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of A 2 k is constructed. It is shown that ( S y l 2 A 2 k ) 2 = S y l 2 ′ A 2 k , k > 2 . It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups C p i , p i ∈ N equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width ( c w ( G ) ) of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups S y l 2 ( A 2 k ) of A 2 k is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group A 2 k , where k > 2 , the permutation group S 2 k , as well as Sylow p-subgroups of S y l 2 A p k as well as S y l 2 S p k ) are equal to 1 was obtained. A commutator width of permutational wreath product B ≀ C n is investigated. An upper bound of the commutator width of permutational wreath product B ≀ C n for an arbitrary group B is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP.

Generation of relative commutator subgroups in Chevalley groups. II

AbstractIn the present paper, which is a direct sequel of our paper [14] joint with Roozbeh Hazrat, we prove an unrelativized version of the standard commutator formula in the setting of Chevalley groups. Namely, let Φ be a reduced irreducible root system of rank ≥ 2, let R be a commutative ring and let I,J be two ideals of R. We consider subgroups of the Chevalley group G(Φ, R) of type Φ over R. The unrelativized elementary subgroup E(Φ, I) of level I is generated (as a group) by the elementary unipotents xα(ξ), α ∈ Φ, ξ ∈ I, of level I. Obviously, in general, E(Φ, I) has no chance to be normal in E(Φ, R); its normal closure in the absolute elementary subgroup E(Φ, R) is denoted by E(Φ, R, I). The main results of [14] implied that the commutator [E(Φ, I), E(Φ, J)] is in fact normal in E(Φ, R). In the present paper we prove an unexpected result, that in fact [E(Φ, I), E(Φ, J)] = [E(Φ, R, I), E(Φ, R, J)]. It follows that the standard commutator formula also holds in the unrelativized form, namely [E(Φ, I), C(Φ, R, J)] = [E(Φ, I), E(Φ, J)], where C(Φ, R, I) is the full congruence subgroup of level I. In particular, E(Φ, I) is normal in C(Φ, R, I).

Finite quotients of braid groups

We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group.

The free spectrum of A5

We obtain a formula for the orders of the relatively free groups of finite rank in the variety generated by the alternating group [Formula: see text]. This is accomplished by explicitly determining the structure of the commutator subgroups of these groups.

Generating Sets and a Structure of the Wreath Product of Groups with Non-Faithful Group Action

Given a permutational wreath product sequence of cyclic groups, we investigate its minimal generating set, the minimal generating set for its commutator and some properties of its commutator subgroup. We generalize the result presented in the book of J. Meldrum [11] also the results of A. Woryna [4]. The quotient group of the restricted and unrestricted wreath product by its commutator is found. The generic sets of commutator of wreath product were investigated. The structure of wreath product with non-faithful group action is investigated. We strengthen the results from the author [17, 19] and construct the minimal generating set for the wreath product of both finite and infinite cyclic groups, in addition to the direct product of such groups. We generalise the results of Meldrum J. [11] about commutator subgroup of wreath products since, as well as considering regular wreath products, we consider those which are not regular (in the sense that the active group A does not have to act faithfully). The commutator of such a group, its minimal generating set and the center of such products has been investigated here. The minimal generating sets for new class of wreath-cyclic geometrical groups and for the commutator of the wreath product are found.

Minimal Generating Set and Structure of the Wreath Product for Groups, Commutator of Wreath Product and the Fundamental Group of the Orbit Morse Function

The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of commutator subgroup of Sylow 2-subgroups of the alternating group ${A_{{2^{k}}}}$ is investigated.It is shown that $(Syl_2 A_{2^k})^2 = Syl'_2 A_{2^k}, \, k>2$.It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N}$ equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. This paper presents upper bounds of the commutator width $(cw(G))$ of a wreath product of groups.A recursive presentation of Sylows $2$-subgroups $Syl_2(A_{{2^{k}}})$ of $A_{{2^{k}}}$ is introduced. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{{2^{k}}}}$, permutation group ${S_{{2^{k}}}}$ and Sylow $p$-subgroups of $Syl_2 A_{p^k}$ ($Syl_2 S_{p^k}$) are equal to 1 is obtained.A commutator width of permutational wreath product $B \wr C_n$ is investigated. An upper bound of the commutator width of permutational wreath product $B \wr C_n$ for an arbitrary group $B$ is found.

ON FINITE-BY-NILPOTENT GROUPS

AbstarctLetγn= [x1,…,xn] be thenth lower central word. Denote byXnthe set ofγn-values in a groupGand suppose that there is a numbermsuch that$|{g^{{X_n}}}| \le m$for eachg∈G. We prove thatγn+1(G)has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

The strict upper bound of ranks of commutator subgroups of finite $p$-groups

The strict upper bound of ranks of commutator subgroups of finite $p$-groups

GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN

If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$. In particular, if $\mathfrak{A}$ is the class of abelian groups, $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$-groups, with special emphasis on the case $\mathfrak{X}=\mathfrak{A}$. Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer $k$.

On Finite Quasi-Core-p p-Groups

Given a positive integer n, a finite group G is called quasi-core-n if ⟨ x ⟩ / ⟨ x ⟩ G has order at most n for any element x in G, where ⟨ x ⟩ G is the normal core of ⟨ x ⟩ in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.

Integrals of groups

An integral of a group G is a group H whose derived group (commutator subgroup) is isomorphic to G. This paper discusses integrals of groups, and in particular questions about which groups have integrals and how big or small those integrals can be. Our main results are: If a finite group has an integral, then it has a finite integral.A precise characterization of the set of natural numbers n for which every group of order n is integrable: these are the cubefree numbers n which do not have prime divisors p and q with q | p − 1.An abelian group of order n has an integral of order at most n1+o(1), but may fail to have an integral of order bounded by cn for constant c.A finite group can be integrated n times (in the class of finite groups) for every n if and only if it is a central product of an abelian group and a perfect group.There are many other results on such topics as centreless groups, groups with composition length 2, and infinite groups. We also include a number of open problems.

A Levi Class Generated by a Quasivariety of Nilpotent Groups

Let L(M) be a class of all groups G in which the normal closure of any element belongs to M; qM is a quasivariety generated by a class M. We consider a quasivariety qH2 generated by a relatively free group in a class of nilpotent groups of class at most 2 with commutator subgroup of exponent 2. It is proved that the Levi class L(qH2) generated by the quasivariety qH2 is contained in the variety of nilpotent groups of class at most 3.

Stable Pseudorepresentations of a Connected Lie Group

The class of the so-called stable pseudorepresentations of groups is introduced, general properties of this class are indicated, and it is shown that every finite-dimensional locally bounded quasirepresentation of a connected Lie group can be approximated by a stable pseudorepresentation which is automatically continuous on the commutator subgroup of the group.