A new characterization of projective special unitary group PSU$(5, q)$

In this paper we prove that a finite group G is isomorphic to the finite projective special unitary group U n ( q ) if and only if they have the same order of Sylow r -normalizer for every prime r .

We prove that each projective special unitary group G can be characterized using only the set of element orders of G and the order of G.

We investigated which elements in some projective special linear and unitary groups are strongly real. In particular we showed every real element in PSL2(q) strongly real. We write a full table for real classes which are not strongly real in the unitary groups as well as a table for the non real classes in the unitary 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.

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).

Article The principal 3-blocks of the 3-dimensional projective special unitary groups in non-defining characteristic was published on September 20, 2001 in the journal Journal für die reine und angewandte Mathematik (volume 2001, issue 539).

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 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.

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.

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).

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.

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 $(n,q)in{(2,7),(2,9),(2,11),(3,3),(4,2)}$‎.

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.

For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, ℤ/nℤ)/±1, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, pv) and the projective special linear groups PSL(2, pv) are realized for allp, and the Suzuki groups Sz(22v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels.

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.