Quasi-Permutation Representations of Metacyclic p-Groups With Non-Cyclic Center

By a quasi-permutation matrix we mean a square matrix over the complex field with non-negative integral trace. Thus every permutation matrix over is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q (G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational field , and let c(G) be the minimal degree of a faithful representation of G by complex quasi-permutation matrices. In this paper we will calculate c(G), q(G), and p(G), where G is a metacyclic p-group with non-cyclic center and p is either 2 or an odd prime number.AMS Subject Classification (2000) 20C15     

Special representations of the group SP(4,q) with minimal degrees

Let G be a finite group. Let p(G) denote the minimal degree of a faithful permutation representation ofG and let q(G) and c(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers, respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. The purpose of this paper is to calculate p(G), q(G), c(G) and r(G) for the group SP(4,q). This revised version was published online in June 2006 with corrections to the Cover Date.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

Minimal degrees of faithful quasi-permutation representations for direct products of p-groups

In [2], the algorithms of c(G), q(G) and p(G), the minimal degrees of faithful quasi-permutation and permutation representations of a finite group G are given. The main purpose of this paper is to consider the relationship between these minimal degrees of non-trivial p-groups H and K with the group H×K.     

Limq → ∞ \frac{c(G)}{r(G)} for Simple Groups

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. For a given finite group G, let c(G) denote the minimal degree of a faithful representation of G by complex quasi-permutation matrices and let r(G) denote the minimal degree of a faithful rational valued character of G. Also let G denote one of the symbols A_l, B_l, C_l, D_l, E₆, E₇, E₈, G₂, F₄, ²B₂, ²E₄, ²G₂, and ³D₄. Let G(q) denote simple group of type G over GF(q). Let c(q) = c(G(q)) and r(q) = r(G(q)). Then we will show that lim Lim_q = 1.     

A note on quasi-permutation representations of groups with two character degrees

In Behravesh (J. Lond. Math. Soc. (2) 55:251–260, 1997), we gave algorithms to calculate c(G), q(G) and p(G) for a finite group G. In this paper we will show that in groups with two character degrees we may have c(G)=q(G)≠p(G).     

On equality of central and class preserving automorphisms of finite p-groups

Let G be a finite non-abelian p-group, where p is a prime. Let Aut_c(G) and Aut_z(G) respectively denote the group of all class preserving and central automorphisms of G. We give a necessary and sufficient condition for G such that Aut_c(G) = Aut_z(G) and classify all finite non-abelian p-groups G with elementary abelian or cyclic center such that Aut_c(G) = Aut_z(G). We also characterize all finite p-groups G of order ≤ p ⁷ such that Aut_z(G) = Aut_z(G) and complete the classification of all finite p-groups of order ≤ p ⁵ for which there exist non-inner class preserving automorphisms.     

Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca-Herz algebras

Let G be a locally compact group, and let R(G) denote the ring of subsets of G generated by the left cosets of open subsets of G. The Cohen-Host idempotent theorem asserts that a set lies in R(G) if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We apply our Cohen-Host type theorems to the study of the Figà-Talamanca-Herz algebras Ap(G) with p∈(1,∞). For arbitrary G, we characterize those closed ideals of Ap(G) that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those G such that Ap(G) is 1-amenable for some—and, equivalently, for all—p∈(1,∞): these are precisely the abelian groups.     

On p-covalenced association schemes

Let (X,S) denote an association scheme where X is a finite set. For a prime p we say that (X,S) is p-covalenced (p-valenced) if every multiplicity (valency, respectively) of (X,S) is a power of p. In the character theory of finite groups Ito's theorem states that a finite group G has a normal abelian p-complement if and only if every character degree of G is a power of p. In this article we generalize Ito's theorem to p-valenced association schemes, i.e., a p-valenced association scheme (X,S) has a normal p-covalenced p-complement if and only if (X,S) is p-covalenced.     

The interior of the Šilov boundary of M(G) is trivial

Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.     

Conjugacy classes of non-normal subgroups of finite p-groups

Let G be a finite p-group, and let ν(G) denote the number of conjugacy classes of non-normal subgroups of G. It is known that either ν(G) ≤ 1 or ν(G) ≥ p. We determine all p-groups G with ν(G) ≤ p + 1.     

Length of Longest Cycles in a Graph Whose Relative Length is at Least Two

Let G be a graph. We denote p(G) and c(G) the order of a longest path and the order of a longest cycle of G, respectively. Let κ(G) be the connectivity of G, and let σ ₃(G) be the minimum degree sum of an independent set of three vertices in G. In this paper, we prove that if G is a 2-connected graph with p(G) ? c(G) ≥ 2, then either (i) c(G) ≥ σ ₃(G) ? 3 or (ii) κ(G)?=?2 and p(G) ≥ σ ₃(G) ? 1. This result implies several known results as corollaries and gives a new lower bound of the circumference.     

Quasi-permutation representations of 2-groups satisfying the Hasse principle

In Behravesh (J Lond Math Soc 55(2):251–260, 1997), c(G), q(G) and p(G) are defined for a finite group G. In this paper, we will calculate c(G), q(G) and p(G) for some 2-groups G satisfying the Hasse principle in Fuma and Ninomiya (Math J Okayama Univ 46:31–38, 2004). We will consider

$G=\langle x, y, z: x^{2^{m-2}}=y^{2}=z^2=1, [x, y]=[y, z]=1, x^{z}=xy \rangle$

where m ≥ 4. By comparing the character tables and Galois conjugacy classes of Irr(G) and Irr(Z(G)), we will show that

$c(G)=q(G)=p(G)= 2c(Z(G))=2^{m-2}+4.$

     

The largest character degree,conjugacy class size and subgroups of finite groups

Let G be a finite group, let b(G) denote the largest irreducible character degree of the group G and let bcl(G) denote the largest conjugacy class size of the group G. We study the relations between the sizes of the nilpotent and solvable subgroups of G and b(G). We also study the relations between the sizes of the nilpotent and solvable subgroups of G and bcl(G).     

Metrizable refinements of group topologies and the pseudo-intersection number

The pseudo-intersection number, denoted p, is the minimum cardinality of a family A⊆P(ω) having the strong finite intersection property but no infinite pseudo-intersection. For every countable topologizable group G, let pG denote the minimum character of a nondiscrete Hausdorff group topology on G which cannot be refined to a nondiscrete metrizable group topology. We show that pG=p.     

Translation-invariant operators continuous in measure

Let G be a compact abelian group and let L(G) be the space of measurable functions on G, equipped with the topology of convergence in measure. The only continuous translation-invariant linear operators on L(G) are the finite linear combinations of the translations themselves.     

On the rational spectra of graphs with abelian singer groups

Let G be a finite abelian group. We investigate those graphs $\text{G}$ admitting G as a sharply 1-transitive automorphism group and all of whose eigenvalues are rational. The study is made via the rational algebra $\text{P}$(G) of rational matrices with rational eigenvalues commuting with the regular matrix representation of G. In comparing the spectra obtainable for graphs in $\text{P}$(G) for various G's, we relate subschemes of a related association scheme, subalgebras of $\text{P}$(G), and the lattice of subgroups of G. One conclusion is that if the order of G is fifth-power-free, any graph with rational eigenvalues admitting G has a cospectral mate admitting the abelian group of the same order with prime-order elementary divisors.     

Long cycles in graphs without hamiltonian paths

For a graph G, p(G) and c(G) denote the order of a longest path and a longest cycle of G, respectively. Bondy and Locke [J.A. Bondy, S.C. Locke, Relative length of paths and cycles in 3-connected graphs, Discrete Math. 33 (1981) 111-122] consider the gap between p(G) and c(G) in 3-connected graphs G. Starting with this result, there are many results appeared in this context, see [H. Enomoto, J. van den Heuvel, A. Kaneko, A. Saito, Relative length of long paths and cycles in graphs with large degree sums, J. Graph Theory 20 (1995) 213-225; M. Lu, H. Liu, F. Tian, Relative length of longest paths and cycles in graphs, Graphs Combin. 23 (2007) 433-443; K. Ozeki, M. Tsugaki, T. Yamashita, On relative length of longest paths and cycles, preprint; I. Schiermeyer, M. Tewes, Longest paths and longest cycles in graphs with large degree sums, Graphs Combin. 18 (2002) 633-643]. In this paper, we investigate graphs G with p(G)−c(G) at most 1 or at most 2, but with no hamiltonian paths. Let G be a 2-connected graph of order n, which has no hamiltonian paths. We show two results as follows: (i) if , then p(G)−c(G)≤1, and (ii) if σ₄(G)≥n+3, then p(G)−c(G)≤2.     

M0(G)-boundaries are M(G)-boundaries

Let M(G) denote the convolution algebra of finite regular complex-valued Borel measures on a locally compact abelian group G, and let M₀(G) be the ideal consisting of those measures whose Fourier-Stieltjes transforms vanish at infinity. Then there is a natural inclusion of the maximal ideal space Δ₀ of M₀(G) in the maximal ideal space of M(G). The main result states that any subset of Δ₀ which is a boundary for M₀(G) is a boundary for M(G). An immediate corollary is that the ?ilov boundary of M₀(G) is dense in the ?ilov boundary of M(G).     

Structure of augmentation quotients of finite homocyclic abelian groups

Let G be a finite abelian group and its Sylow p-subgroup a direct product of copies of a cyclic group of order p~r,i.e.,a finite homocyclic abelian group.LetΔ~n (G) denote the n-th power of the augmentation idealΔ(G) of the integral group ring ZG.The paper gives an explicit structure of the consecutive quotient group Q_n(G)=Δ~n(G)/Δ~(n 1)(G) for any natural number n and as a consequence settles a problem of Karpilovsky for this particular class of finite abelian groups.