Commutator Subgroup Research Articles

The group generated by Riordan involutions

We prove that any element in the group generated by the Riordan involutions is the product of at most four of them. We also give a description of this subgroup as a semidirect product of a special subgroup of the commutator subgroup and the Klein four-group.

Group rings that are UJ rings

The set of all elements r of a ring R such that is a unit for every unit u extends the Jacobson radical J(R). R is a UJ ring (ΔU ring, respectively) if its units are of the form ( respectively). Using a local characterization of ΔU rings, we describe structure of group rings that are UJ rings; if RG is a UJ group ring, then R is a UJ ring, G is a 2-group and, for every nontrivial finitely generated subgroup H of G, the commutator subgroup of H is proper subgroup of H. Conversely, if R is a UJ ring and G a locally finite 2-group, then RG is a UJ ring. In particular, if G is solvable, RG is a UJ ring if and only if R is UJ and G is a 2-group.

Crystallographic groups and flat manifolds from surface braid groups

Let M be a compact surface without boundary, and n≥2. We analyse the quotient group Bn(M)/Γ2(Pn(M)) of the surface braid group Bn(M) by the commutator subgroup Γ2(Pn(M)) of the pure braid group Pn(M). If M is different from the 2-sphere S2, we prove that Bn(M)/Γ2(Pn(M))≅Pn(M)/Γ2(Pn(M))⋊φSn, and that Bn(M)/Γ2(Pn(M)) is a crystallographic group if and only if M is orientable.If M is orientable, we prove a number of results regarding the structure of Bn(M)/Γ2(Pn(M)). We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of Bn(M)/Γ2(Pn(M)) isomorphic either to Sn or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection Bn(M)/Γ2(Pn(M))⟶Sn is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups G˜n,g of Bn(M)/Γ2(Pn(M)) of dimension 2ng and whose holonomy group is the finite cyclic group of order n, and if Xn,g is a flat manifold whose fundamental group is G˜n,g, we prove that it is an orientable Kähler manifold that admits Anosov diffeomorphisms.

Un-Weyl-ing the Clifford Hierarchy

The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being the ubiquitous Pauli group, via the Weyl (i.e., Pauli) expansion of unitaries at these levels. In particular, we characterize the support of the standard Clifford operations on the Pauli group. Since conjugation of a Pauli by a third level unitary produces traceless Hermitian Cliffords, we characterize their Pauli support as well. Semi-Clifford unitaries are known to have ancilla savings in the teleportation model, and we explore their Pauli support via symplectic transvections. Finally, we show that, up to multiplication by a Clifford, every third level unitary commutes with at least one Pauli matrix. This can be used inductively to show that, up to a multiplication by a Clifford, every third level unitary is supported on a maximal commutative subgroup of the Pauli group. Additionally, it can be easily seen that the latter implies the generalized semi-Clifford conjecture, proven by Beigi and Shor (2010). We discuss potential applications in quantum error correction and the design of flag gadgets.

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial ℤ-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial ℤ-groups, we finally obtain the structure and invariants of the group G.

On finite-by-nilpotent profinite groups

Let $gamma_n=[x_1,ldots,x_n]$ be the $n$th lower central word‎. ‎Suppose that $G$ is a profinite group‎ ‎where the conjugacy classes $x^{gamma_n(G)}$ contains less than $2^{aleph_0}$‎ ‎elements‎ ‎for any $x in G$‎. ‎We prove that then $gamma_{n+1}(G)$ has finite order‎. ‎This generalizes the much celebrated‎ ‎theorem of B‎. ‎H‎. ‎Neumann that says that the commutator subgroup of a BFC-group is finite‎. ‎Moreover‎, ‎it implies that‎ ‎a profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that‎ ‎$x^{gamma_n(G)}$ contains less than $2^{aleph_0}$‎ ‎elements‎, ‎for any $xin G$‎.

Commutator subgroups of generalized Hecke and extended generalized Hecke groups,II

Let $p_{1},\ \cdots,\ p_{n}$ be integers where $n\geq 2$ and each $p_{i} \geq 2$. Let also $H(p_{1},\ \cdots,\ p_{n})$ be the generalized Hecke group associated to all $p_{i}\geq 2.$ In this paper, we study the commutator subgroups $H^{\prime}(p_{1},\ \cdots,\ p_{n})$ and $\overline{H}^{\prime }(p_{1},\ \cdots,\ p_{n})$ of the generalized Hecke group $H(p_{1},\ \cdots,\ p_{n})$ and the extended generalized Hecke group $\overline{H}% (p_{1},\ \cdots,\ p_{n})$. We give the generators and the signatures of $% H^{\prime}(p_{1},\ \cdots,\ p_{n})$ and $\overline{H}^{\prime}(p_{1},\ \cdots,\ p_{n})$.

Automatic realization of Hopf Galois structures

We consider Hopf Galois structures on a separable field extension [Formula: see text] of degree [Formula: see text], for [Formula: see text] an odd prime number, [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] has at most one abelian type of Hopf Galois structures. For a nonabelian group [Formula: see text] of order [Formula: see text], with commutator subgroup of order [Formula: see text], we prove that if [Formula: see text] has a Hopf Galois structure of type [Formula: see text], then it has a Hopf Galois structure of type [Formula: see text], where [Formula: see text] is an abelian group of order [Formula: see text] and having the same number of elements of order [Formula: see text] as [Formula: see text], for [Formula: see text].

Commutators of Relative and Unrelative Elementary Groups, Revisited

Let R be any associative ring with 1, let n ≥ 3, and let A,B be two-sided ideals of R. In the present paper, we show that the mixed commutator subgroup [E(n,R,A),E(n,R,B)] is generated as a group by the elements of the two following forms: 1) zij(ab, c) and zij (ba, c), 2) [tij(a), tji(b)], where 1 ≤ i ≠ j ≤ n, a ∈ A, b ∈ B, c ∈ R. Moreover, for the second type of generators, it suffices to fix one pair of indices (i, j). This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality [E(n,R,A),E(n,R,B)] = [E(n,A),E(n,B)], and many further corollaries can be derived for rings subject to commutativity conditions. Bibliography: 36 titles.

Commutators and commutator subgroups of finite p-groups

We present a characterisation, up to isoclinism, of finite p-groups G with γ2(G), the commutator subgroup of G, of order p4 and exponent p such that not all elements of γ2(G) are commutators.

Kerdock Codes Determine Unitary 2-Designs

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits.

The adjoint group of a Coxeter quandle

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles of $W$ corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group of $Q_W$ is isomorphic to the commutator subgroup of $W$. Finally, the root system $\Phi_W$ associated with a Coxeter group $W$ turns out to be a rack. We prove that the adjoint group of $\Phi_W$ is isomorphic to the adjoint group of $Q_W$.

Invariant Radon measures and minimal sets for subgroups of [formula omitted

Let G be a subgroup of Homeo+(R) without crossed elements. We show the equivalence among three items: (1) existence of G-invariant Radon measures on R; (2) existence of minimal closed subsets of R; (3) nonexistence of infinite towers covering the whole line. For a nilpotent subgroup G of Homeo+(R), we show that G always has an invariant Radon measure and a minimal closed set if every element of G is C1+α(α>0); a counterexample of C1 commutative subgroup of Homeo+(R) is constructed.

The twisted commutator subgroups and questions of Fel’shtyn and Troitsky

In this paper, we study the notion of the twisted conjugacy separability developed in Fel’shtyn and Troitsky (J K-Theory 1:1–40, 2008). We give an example of group with twisted conjugacy separability, but without residual finiteness. Thus, we find the negative answers on the several open questions formulated by Fel’shtyn and Troitsky (Topol Appl 157:1724–1735, 2010). In addition, we discuss the generalization of usual the commutator subgroup, especially for some relations among Nielsen fixed point theory and the twisted commutator subgroup.

The Axiomatic Rank of the Levi Class Generated by the Almost Abelian Quasivariety of Nilpotent Groups

Let $$\mathcal{M}$$ be a quasivariety generated by a relatively free group in a class of nilpotent groups of step $$\leq 2$$ with commutator subgroups of prime exponent $$p,$$ $$p\neq 2.$$ A class $$L(\mathcal{M})$$ of all groups $$G$$ the normal closure of any element in which belongs to $$\mathcal{M}$$ is called the Levi class generated by $$\mathcal{M}.$$ It is proved that $$L(\mathcal{M})$$ has finite axiomatic rank, i.e., $$L(\mathcal{M})$$ can be defined by a system of quasi-identities with a finite number of variables.

On nilpotency of higher commutator subgroups of a finite soluble group

Let G be a finite soluble group and $$G^{(k)}$$ the kth term of the derived series of G. We prove that $$G^{(k)}$$ is nilpotent if and only if $$|ab|=|a||b|$$ for any $$\delta _k$$ -values $$a,b\in G$$ of coprime orders. In the course of the proof, we establish the following result of independent interest: let P be a Sylow p-subgroup of G. Then $$P\cap G^{(k)}$$ is generated by $$\delta _k$$ -values contained in P (Lemma 2.5). This is related to the so-called focal subgroup theorem.

Topological full groups and t.d.l.c. completions of Thompson’s V

We show that every topological full group coming from a one-sided irreducible shift of finite type, in the sense of Matui, is isomorphic to a group of tree almost automorphisms preserving an edge colouring of the tree. As an application, we prove that it admits totally disconnected, locally compact completions of arbitrary nonempty finite local prime content. The completions we construct have open, simple commutator subgroup. Our results apply to Thompson’s V, providing answers to two questions asked by Le Boudec and Wesolek.

On commutator subgroups of Sylow 2-subgroups of the alternating group, and the commutator width in wreath products

We find the size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group and investigate the structure of the commutator subgroup of Sylow 2-subgroups of the alternating group $${A_{{2^{k}}}}$$ . This work continues previous investigations of the author (Skuratovskii in Cybern Syst Anal 45(1):25–37, 2009; ROMAI J 13(1):117–139, 2017), where minimal generating sets of Sylow 2-subgroups of alternating groups were found.

Strong conciseness of coprime and anti-coprime commutators

A coprime commutator in a profinite group G is an element of the form [x, y], where x and y have coprime order and an anti-coprime commutator is a commutator [x, y] such that the orders of x and y are divisible by the same primes. In the present paper, we establish that a profinite group G is finite-by-pronilpotent if the cardinality of the set of coprime commutators in G is less than $$2^{\aleph _0}$$ . Moreover, a profinite group G has finite commutator subgroup $$G'$$ if the cardinality of the set of anti-coprime commutators in G is less than $$2^{\aleph _0}$$ .

Character tables and defect groups

Let B be a block of a finite group G with defect group D. We prove that the exponent of the center of D is determined by the character table of G. In particular, we show that D is cyclic if and only if B contains a “large” family of irreducible p-conjugate characters. More generally, for abelian D we obtain an explicit formula for the exponent of D in terms of character values. In small cases even the isomorphism type of D is determined in this situation. Moreover, it can read off from the character table whether |D/D′|=4 where D′ denotes the commutator subgroup of D. We also propose a new characterization of nilpotent blocks in terms of the character table.