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

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.

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.

We show that projective unitary groups, or more generally, central quotients of products of unitary groups are totally $N$-determined when considered as $p$-compact groups at odd primes $p$.

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 F be any field of prime characteristic p and let q be a power of p. We assume that F contains the finite field of order q. A q-polynomial L over F is an element of the polynomial ring F[x] with the property that all exponents are a power of q. We assume that the coefficient of the x term of L is nonzero. We investigate the Galois group G of L over F, under the assumption that L(x)/x is irreducible in F[x]. It is well known that if L has q-degree n, its roots are an n-dimensional vector space over the field of order q and G acts linearly on this space.Our main theorem is the following. We consider a monic q-polynomial L whose q-degree is a prime, r, say, with L(x)/x irreducible over F. Assuming that the coefficient of the x term of L is (−1)r, we show that the Galois group of L over F must be the special linear group SL(r,q) of degree r over the field of order q when q>2. We also prove a projective version for the projective special linear group PSL(r,q), and we present the analysis when q=2.

The fundamental coupling constants of physics in connection with the dimension of the special orthogonal and unitary groups

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).

Maximal Overgroups of Singer Elements in Classical Groups

In the example of the nonlinear Klein-Gordon equation, we demonstrate that even-indexed hypergeometric solutions admit matrix representations that can be associated with special unitary groups. For the index 2 in particular, this correspondence is shown to be1 :1. For the odd index 3, we show that no anticommuting matrices exist in the class of unitary anti-Hermitian matrices. We also show that the solutions obtained describe going around the potential barrier in the electron-proton transport problem.

Let A be a linear transformation with minimum polynomial mA(t) and characteristic polynomial cA(t). We say that A is cyclic if mA(t) = cA(t), semisimple if mA(t) has no repeated factors, and separable if it is both cyclic and semisimple. The proportions of elements of GL(d, q) which are cyclic or separable have been investigated by, amongst others, Neumann and Praeger [7], Wall [10], and Fulman [4] who also studies semisimple elements. Making use of their results, we gave estimates in [1] for these proportions in SL(d, q). Fulman, Neumann and Praeger [5] have recently found good estimates for the corresponding proportions in the finite classical groups, U(d, q2), Sp(2m, q) and O±(d, q). They raise the question of whether similar methods may be used to calculate the proportions in various subgroups and supergroups of the classical groups. In a series of three papers, we shall show how this may be done. In this, the first paper, we deal with the special unitary groups SU(d, q2), consisting of unitary transformations with determinant 1. In the second paper [2] we shall focus on the conformal groups GSp(2m, q) and GO±(d, q), and in the third paper [3] we shall consider how the method may be extended to a number of groups of orthogonal type, including the special orthogonal group and the derived group of the orthogonal group. Our method in [1] involved the vector space cycle index of Kung [6] and Stong [8], which we describe briefly in Section 2. Similarly, in the current series

A submanifold of a Riemannian manifold is called reflective, if it is a connected component of an involutive isometry.\nIf every shortest geodesic arc of a complete submanifold is still shortest in the ambient space, we say that the submanifold is convex.\nIn this note we show that reflective submanifolds in special unitary groups are convex.

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.

We give a generalisation of Deligne–Lusztig varieties for general and special linear groups over finite quotients of the ring of integers in a non-archimedean local field. Previously, a generalisation was given by Lusztig by attaching certain varieties to unramified maximal tori inside Borel subgroups. In this paper we associate a family of so-called extended Deligne–Lusztig varieties to all tamely ramified maximal tori of the group. Moreover, we analyse the structure of various generalised Deligne–Lusztig varieties, and show that the “unramified” varieties, including a certain natural generalisation, do not produce all the irreducible representations in general. On the other hand, we prove results which together with some computations of Lusztig show that for SL 2 ( F q 〚 ϖ 〛 / ( ϖ 2 ) ) , with odd q , the extended Deligne–Lusztig varieties do indeed afford all the irreducible representations.