Projective Unitary Group Research Articles

The involution module of $\mathrm{PSU}_{3}(2^{2f})$

For any group $G$ the involutions $\mathcal{I}$ in $G$ form a $G$-set under conjugation. The corresponding $kG$-permutation module $k\mathcal{I}$ is known as the involution module of $G$, with $k$ an algebraically closed field of characteristic two. In this paper we discuss the involution module of the projective special unitary group $\mathrm{PSU}_{3}(4^{f})$.

Characterization of the Critical Sets of Quantum Unitary Control Landscapes

This work considers various families of quantum control landscapes (i.e. objective functions for optimal control) for obtaining target unitary transformations as the general solution of the controlled Schrodinger equation. We examine the critical point structure of the kinematic landscapes J F (U) = ||(U - W)A|| 2 and J P (U) = ||A|| 4 - |Tr(AA † W † U)| 2 defined on the unitary group U(H) of a finite-dimensional Hilbert space H. The parameter operator A E (H) is allowed to be completely arbitrary, yielding an objective function that measures the difference in the actions of U and the target W on a subspace of state space, namely the column space of A. The analysis of this function includes a description of the structure of the critical sets of these kinematic landscapes and characterization of the critical points as maxima, minima, and saddles. In addition, we consider the question of whether these landscapes are Morse-Bott functions on U(H). Landscapes based on the intrinsic (geodesic) distance on U(H) and the projective unitary group PU(H) are also considered. These results are then used to deduce properties of the critical set of the corresponding dynamical landscapes.

Minimal logarithmic signatures for the unitary group $$U_n(q)$$ U n ( q )

As a special type of factorization of finite groups, logarithmic signature (LS) is used as one of the main components of the private key cryptosystem $$PGM$$PGM and the public key cryptosystems $$MST_1$$MST1, $$MST_2$$MST2 and $$MST_3$$MST3. An LS with the shortest length is called a minimal logarithmic signature (MLS) and is even desirable for cryptographic constructions. The MLS conjecture states that every finite simple group has an MLS. Recently, Singhi et al. proved that the MLS conjecture is true for some families of simple groups. In this paper, we prove the existence of MLSs for the unitary group $$U_n(q)$$Un(q) and construct MLSs for a type of simple groups--the projective special unitary group $$PSU_n(q)$$PSUn(q).

The period-index problem for twisted topologicalK–theory

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of dimension d, we give upper bounds on the index depending only on d and the order of the class. By the Oka principle, this also solves the period-index problem for the analytic Brauer group of any Stein space that has the homotopy type of a finite CW-complex. Our methods use twisted topological K-theory, which was first introduced by Donovan and Karoubi. We also study the cohomology of the projective unitary groups to give cohomological obstructions to a class being represented by an Azumaya algebra of degree n. Applying this to the finite skeleta of the Eilenberg-MacLane space K(Z/l,2), where l is a prime, we construct a sequence of spaces with an order l class in Br, but whose indices tend to infinity.

The Projective Special Unitary Group [formula omitted] and their Codes

In this paper, all the binary codes from the primitive permutation group PSU2(16) are constructed. It is shown that the groups PSU2(16) : 4 and S17 are the automorphism groups of the constructed codes.

The topological period-index problem over 6-complexes

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah–Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and solve the topological version of the period–index problem in full for finite CW complexes of dimension less than 6. Conditions are established that, if they were met in the cohomology of a smooth complex 3-fold variety, would disprove the ordinary period–index conjecture. Examples of higher-dimensional varieties meeting these conditions are provided. We use our results to furnish an obstruction to realizing a period-2 Brauer class as the class associated to a sheaf of Clifford algebras, and varieties are constructed for which the total Clifford invariant map is not surjective. No such examples were previously known.

A CLOSER LOOK AT QUANTUM CONTROL LANDSCAPES AND THEIR IMPLICATION FOR CONTROL OPTIMIZATION

The control landscape for various canonical quantum control problems is considered. For the class of pure-state transfer problems, analysis of the fidelity as a functional over the unitary group reveals no suboptimal attractive critical points (traps). For the actual optimization problem over controls in L2(0, T), however, there are critical points for which the fidelity can assume any value in (0, 1), critical points for which the second order analysis is inconclusive, and traps. For the class of unitary operator optimization problems analysis of the fidelity over the unitary group shows that while there are no traps over U(N), traps already emerge when the domain is restricted to the special unitary group. The traps on the group can be eliminated by modifying the performance index, corresponding to optimization over the projective unitary group. However, again, the set of critical points for the actual optimization problem for controls in L2(0, T) is larger and includes traps, some of which remain traps even when the target time is allowed to vary.

A Characterization of Projective Special Unitary Group U3(7) by nse

Let G a group and ω(G) be the set of element orders of G. Let k∈ω(G) and let sk be the number of elements of order k in G. Let nse(G)={sk∣k∈ω(G)}. In Khatami et al. and Liu's works, L3(2) and L3(4) are uniquely determined by nse(G). In this paper, we prove that if G is a group such that nse(G) = nse(U3(7)), then G≅U3(7).

A characterization of projective special unitary group U3(5) by nse

Let G be a group and ω(G) be the set of element orders of G. Let k∈ω(G) and sk be the number of elements of order k in G. Let nse(G)={sk∣k∈ω(G)}. In Khatami et al. and Liu’s works L3(2) and L3(4) are unique determined by nse(G). In this paper, we prove that if G is a group such that nse(G)=nse(U3(5)), then G≅U3(5).

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p 1, p 2, p 3 be primes. This is the second article in a series of three on the (p 1, p 2, p 3)-generation of the finite projective special unitary and linear groups PSU3(p n ), PSL3(p n ), where we say a noncyclic group is (p 1, p 2, p 3)-generated if it is a homomorphic image of the triangle group T p 1, p 2, p 3 . This paper is concerned with the case where p 1 = 2 and p 2 = p 3. We determine for any prime p 2 the prime powers p n such that PSU3(p n ) (respectively, PSL3(p n )) is a quotient of T = T 2, p 2, p 2 . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU3(p n )) (respectively, Hom(T, PSL3(p n ))) is surjective as p n tends to infinity.

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Let p1, p2, p3 be primes. This is the final paper in a series of three on the (p1, p2, p3)-generation of the finite projective special unitary and linear groups PSU 3(pn), PSL 3(pn), where we say a noncyclic group is (p1, p2, p3)-generated if it is a homomorphic image of the triangle group Tp1, p2, p3 . This article is concerned with the case where p1 = 2 and p2 ≠ p3. We determine for any primes p2 ≠ p3 the prime powers pn such that PSU 3(pn) (respectively, PSL 3(pn)) is a quotient of T = T2, p2, p3 . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU 3(pn)) (respectively, Hom(T, PSL 3(pn))) is surjective as pn tends to infinity.

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups I

Let p 1, p 2, p 3 be primes. This is the first in a series of three articles concerned with the (p 1, p 2, p 3)-generation of the projective special unitary and linear groups PSU3(p n ), PSL3(p n ), where we say a noncyclic group is (p 1, p 2, p 3)-generated if it is a quotient of the triangle group T p 1, p 2, p 3 . We present our results which are probabilistic, asymptotic and also deterministic, and provide the machinery needed to prove them when p 1, p 2, p 3 are distinct or all odd. Complete proofs are given when p 1, p 2, p 3 are odd. The second and final article investigate the cases where p 1 = 2 and p 2 = p 3, and p 1 = 2 and p 2 ≠ p 3, respectively.

Hermitian Morita Theory and Unitary Groups of Hyperbolic Quadratic Modules over Maximal Orders in Central Simple Algebras

Let R = Z with quotient field say K. Let Λ and Λ1 be maximal R-orders with antistructure in central simple K-algebras A = M n (D) and A 1 = M n (D) where D and D 1 are separable over K. Let (M, h) and (N, k) be hyperbolic quadratic, hermitian Λ and Λ1 modules, respectively. Let be an arbitrary isomorphism of the corresponding projective unitary groups. Under some minimal conditions there is a category equivalence from the category of hermitian Λ-modules to that of hermitian Λ1-modules which arises fronm a hermitian Morita context and produces Φ by its action on morphisms.

RECOGNIZING SOME FINITE SIMPLE GROUPS BY NONCOMMUTING GRAPH

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the noncentral elements of G, and two distinct vertices x and y of ∇(G) are joined by an edge if and only if x and y do not commute as elements of G, i.e. [x, y] ≠ 1. The finite group L is said to be recognizable by noncommuting graph if, for every finite group G, ∇(G) ≅ ∇ (L) implies G ≅ L. In the present article, it is shown that the noncommuting graph of a group with trivial center can determine its prime graph. From this, the following theorem is derived. If two finite groups with trivial centers have isomorphic noncommuting graphs, then their prime graphs coincide. It is also proved that the projective special unitary groups U4(4), U4(8), U4(9), U4(11), U4(13), U4(16), U4(17) and the projective special linear groups L9(2), L16(2) are recognizable by noncommuting graph.

Unitary grassmannians

We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.

Symmetric designs and projective special unitary groups $text{PSU}_{5}(q)$

In this article‎, ‎we prove that if a nontrivial symmetric $(v‎, ‎k‎, ‎lambda)$ design admit a flag-transitive and point-primitive automorphism group $G$‎, ‎then the socle $X$ of $G$ cannot be a projective special unitary group of dimension five‎. ‎As a corollary‎, ‎we list all exist nineteen non-isomorphism such designs in which $lambdain{1,2,3,4,6,12‎, ‎16‎, ‎18}$ and $X=text{PSU}_n(q)$ with $(n,q)in{(2,7),(2,9),(2,11),(3,3),(4,2)}$‎.

Commuting Involution Graphs for $3$-Dimensional Unitary Groups

For a group $G$ and $X$ a subset of $G$ the commuting graph of $G$ on $X$, denoted by $\cal{C}(G,X)$, is the graph whose vertex set is $X$ with $x,y\in X$ joined by an edge if $x\neq y$ and $x$ and $y$ commute. If the elements in $X$ are involutions, then $\cal{C}(G,X)$ is called a commuting involution graph. This paper studies $\cal{C}(G,X)$ when $G$ is a 3-dimensional projective special unitary group and $X$ a $G$-conjugacy class of involutions, determining the diameters and structure of the discs of these graphs.

Simplicity of the projective unitary group of the multiplier algebra of a simple stable nuclear $C*$-algebra

Simplicity of the projective unitary group of the multiplier algebra of a simple stable nuclear $C*$-algebra

OD-characterization of almost simple groups related to U 3(5)

Let G be a finite group with order |G| = p1α1p2α2 … pkαk, where p1 < p2 < … < pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg-Kegel graph) denoted by Γ(G) (or GK(G)). This graph is constructed as follows: The vertex set of it is π(G) = {p1, p2, …, pk} and two vertices pi, pj with i ≠ j are adjacent by an edge (and we write pi ∼ pj) if and only if G contains an element of order pipj. The degree deg(pi) of a vertex pi ∈ π(G) is the number of edges incident on pi. We define D(G):= (deg(p1), deg(p2), …, deg(pk)), which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k nonisomorphic groups H such that |H| = |G| and D(H) = D(G). Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L:= U3(5) be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.

A NEW CHARACTERIZATION OF U4(7) BY ITS NONCOMMUTING GRAPH

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).