Modular invariant models of leptons at level 7

We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ7, which is isomorphic to PSL(2, Z7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL2(7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

Centralizers in a group whose central factor is simple

In this paper, we find the number of the element centralizers of a finite group [Formula: see text] such that the central factor of [Formula: see text] is the projective special linear group of degree 2 or the Suzuki group. Our results generalize some main results of [Ashrafi and Taeri, On finite groups with a certain number of centralizers, J. Appl. Math. Comput. 17 (2005) 217–227; Schmidt, Zentralisatorverbände endlicher Gruppen, Rend. Sem. Mat. Univ. Padova 44 (1970) 97–131; Zarrin, On element centralizers in finite groups, Arch. Math. 93 (2009) 497–503]. Also, we give an application of these results.

Computing of Z- valued Characters for the Projective Special Linear Group L2 (2m) and the Conway Group Co3

<p><span lang="EN-US">According to the main result of W. Feit and G. M. Seitz (see, Illinois J. Math. 33 (1), 103-131, 1988), the projective special linear group L₂ (2^m) for m = 3, 4, 5 and the smallest Conway group Co₃ are unmatured groups. In this paper, we continue our study on special finite groups (see Int. J. Theo. Physics, Group Theory, and Nonlinear Optics (17)1, 57-62, 2013) and the dominant classes and Q- conjugacy characters for the above groups are derived.</span></p>

THE GROUP OF BI-GALOIS OBJECTS OVER THE COORDINATE ALGEBRA OF THE FROBENIUS–LUSZTIG KERNEL OF SL(2)

We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over uq(sl(n))*, the coordinate algebra of the Frobenius–Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2.

Erdös-Ko-Rado theorem in some linear groups and some projective special linear group

Let V be the 2-dimensional column vector space over a finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document} (where q is necessarily a power of a prime number) and let ℙq be the projective line over \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. In this paper, it is shown that GL2(q), for q ≠ 3, and SL2(q) acting on V − {0} have the strict EKR property and GL2(3) has the EKR property, but it does not have the strict EKR property. Also, we show that GLn(q) acting on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left( {\mathbb{F}_q } \right)^n - \left\{ 0 \right\}$$ \end{document} has the EKR property and the derangement graph of PSL2(q) acting on ℙq, where q ≡ −1 (mod 4), has a clique of size q + 1.

On recognition by prime graph of the projective special linear group over GF(3)

Let G be a finite group. The prime graph of G is denoted by ?(G). We prove that the simple group PSLn(3), where n ? 9, is quasirecognizable by prime graph; i.e., if G is a finite group such that ?(G) = ?(PSLn(3)), then G has a unique nonabelian composition factor isomorphic to PSLn(3). Darafsheh proved in 2010 that if p > 3 is a prime number, then the projective special linear group PSLp(3) is at most 2-recognizable by spectrum. As a consequence of our result we prove that if n ? 9, then PSLn(3) is at most 2-recognizable by spectrum.