Projective unitary groups are totally $N$-determined $p$-compact groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups.

There exists an infinite family of p p -compact groups whose Weyl groups correspond to the finite p p -adic pseudoreflection groups G ( q , r , n ) G(q,r,n) of family 2a in the Clark-Ewing list. In this paper we study these p p -compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical J J -homomorphism. Finally, we also describe a faithful complexification homomorphism from these p p -compact groups to the p p -completion of unitary compact Lie groups.

We provide descriptions for the moduli spaces Rep(Γ, PU(m)), where Γ is any finitely generated Abelian group and PU(m) is the group of m × m projective unitary matrices. As an application, we show that for any connected CW-complex X with π1(X)≅Zn, the natural map π0(Rep(π1(X), PU(m))) → [X, BPU(m)] is injective, hence providing a complete enumeration of the isomorphism classes of flat principal PU(m)-bundles over X.

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.

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).

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.

Let G be a finite group and cd(G) be the set of all irreducible complex character degrees of G without multiplicities. The aim of this paper is to propose an extension of Huppert’s conjecture from non-Abelian simple groups to almost simple groups of Lie type. Indeed, we conjecture that if H is an almost simple group of Lie type with cd(G)=cd(H), then there exists an Abelian normal subgroup A of G such that G/A≅H. It is furthermore shown that G is not necessarily the direct product of H and A. In view of Huppert’s conjecture, we also show that the converse implication does not necessarily hold for almost simple groups. Finally, in support of this conjecture, we will confirm it for projective general linear and unitary groups of dimension 3.

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

On the group with the same orders of Sylow normalizers as the finite projective special unitary group

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).

The concepts of normalized irreducible tensorial matrices (NITM) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M(α1 × α2). Elements [X] ∈ M(α1 α2) transform under Ga ∈ G according to [Ga] [X][G−1a], where [Ga] and [G−1a] belong to irreducible representations of G. The usual properties of NITM and the Wigner–Eckart theorem follow from these results, which are valid for both finite and compact unitary groups. The NITM span M(α1 × α2) are orthonormal under the trace and transform irreducibly with respect to G. This NITM basis of M(α1 × α2) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffith's V coefficients for the octahedral group are tabulated and used to construct a six‐coordinate superposition Hamiltonian.

Optimization problems over compact Lie groups have been extensively studied due to their broad applications in linear programming and optimal control. This paper analyzes least square problems over a noncompact Lie group, the symplectic group $\Sp(2N,\R)$, which can be used to assess the optimality of control over dynamical transformations in classical mechanics and quantum optics. The critical topology for minimizing the Frobenius distance from a target symplectic transformation is solved. It is shown that the critical points include a unique local minimum and a number of saddle points. The topology is more complicated than those of previously studied problems on compact Lie groups such as the orthogonal and unitary groups because the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group brings significant nonlinearity to the problem. Nonetheless, the lack of traps guarantees the global convergence of local optimization algorithms.

We study the complex-time Segal-Bargmann transform Bs,τKN on a compact type Lie group KN, where KN is one of the following classical matrix Lie groups: the special orthogonal group SO(N,R), the special unitary group SU(N), or the compact symplectic group Sp(N). Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group U(N). We provide an effective method of computing the action of the Segal-Bargmann transform on trace polynomials, which comprise a subspace of smooth functions on KN extending the polynomial functional calculus. Using these results, we show that as N→∞, the finite-dimensional transform Bs,τKN has a meaningful limit Gs,τ which can be identified as an operator on the space of complex Laurent polynomials.

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

Bounded normal generation and invariant automatic continuity