Regular Orbits of Extra-special Groups

2016 ◽  
Vol 23 (04) ◽  
pp. 681-688
Author(s):  
Joshua E. Goodson

In this paper, for odd primes p we find a recursive formula for the number of regular orbits of an extra-special p-group of exponent p2 acting on a faithful irreducible module over a finite field. This complements an earlier result of Foulser for extra-special groups of order p.

10.37236/7008 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Yuta Watanabe

In this paper, we introduce an algebra $\mathcal{H}$ from a subspace lattice with respect to a fixed flag which contains its incidence algebra as a proper subalgebra. We then establish a relation between the algebra $\mathcal{H}$ and the quantum affine algebra $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$, where $q$ denotes the cardinality of the base field. It is an extension of the well-known relation between the incidence algebra of a subspace lattice and the quantum algebra $U_{q^{1/2}}(\mathfrak{sl}_2)$. We show that there exists an algebra homomorphism from $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$ to $\mathcal{H}$ and that any irreducible module for $\mathcal{H}$ is irreducible as an $U_{q^{1/2}}(\widehat{\mathfrak{sl}}_2)$-module.


2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing

Let \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} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given 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}. For any \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} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \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} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \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} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.


Author(s):  
Maulana Akbar Shah @ U Tun Aung

Abstract In addition to compliance with the five pillars of Islam, Da’wah Islamiyyah is an indispensable act for Muslims who view their life missions as to propagate Islam. The act of da’wah is a mission only undertaken by those who are selected to do so, as Allah SWT said: “You are the best of peoples, taken out for mankind, you enjoin what is right, forbid what is wrong…,” (Al-Qur’an, Al- `Imran, 3:110). The phrase “you are taken out for mankind” is a very clear phrase which refers to a special group. Many du’at propagate Islam through regular procedures. In this contemporary day and age, it would be more successful if creative da’wah methodologies are used to special groups where special talent is required. Examples of these special target groups are individuals such as [1] pregnant mothers, [2] mothers of new born babies, [3] parents of toddlers, [4] parents of children [5] youth, teenagers and adolescents, [6] married couples, [7] elderly or aged individuals and [8] non-Muslims.  They are good targets to promote Islam. Thus, du’at must have special talents and knowledge to do so. In this regard, du’at must be equipped with exceptional knowledge of the target groups to propagate Islam effectively. The present paper attempts to highlight the distinctive natures of the above mentioned categories of individuals and explores how du’at should approach them when he or she encounters them for the purpose of promoting Islam.        Keywords: Call to Islam, Da’wah Islamiyyah, creative da’wah, creative da’ie, enjoin good, forbid evil.   Abstrak Selain mematuhi 5 rukun Islam, dakwah Islamiyyah tidak dapat diasingkan daripada seseorang Muslim yang bermatlamat untuk menyebarkan ajaran Islam. Berdakwah adalah suatu tugas yang hanya dijalankan oleh mereka yang terpilih. Seperti yang diwahyukan oleh Allah SWT “Kamu adalah sebaik-baik umat yang dilahirkan bagi umat manusia, kerana kamu menyuruh berbuat segala perkara yang baik dan melarang daripada segala perkara yang buruk dan keji...” (Al-Quran, Al-Imran, 3:110). Frasa “umat yang dilahirkan bagi umat manusia”  jelas dinyatakan di mana ia merujuk kepada satu golongan yang istimewa. Kebanyakan pendakwah menyebarkan Islam melalui metodologi yang sudah menjadi kebiasaan. Bagaimanapun, pada zaman ini, pendakwahan akan lebih berjaya jika kaedah yang kreatif digunakan kepada beberapa golongan yang tertentu. Antara contoh golongan yang dimaksudkan adalah [1] wanita yang mengandung, [2] ibu kepada bayi yang baru lahir, [3] ibu bapa kepada anak-anak kecil, [4] ibu bapa kepada kanak-kanak, [5] golongan belia dan remaja, [6] pasangan suami isteri, [7] warga emas, dan [8] golongan bukan Islam. Mereka merupakan sasaran yang sempurna untuk mempelajari Islam. Oleh hal demikian, para pendakwah mesti mempunyai bakat yang istimewa dan ilmu yang mencukupi untuk berdakwah. Para pendakwah juga harus memahami kumpulan sasaran dengan menyeluruh supaya penyebaran ajaran Islam akan lebih berkesan. Tujuan penulisan ini adalah untuk menggariskan sifat unik golongan yang dinyatakan di atas dan menerokai bagaimana para pendakwah harus mendekati mereka dengan niat untuk menyebarkan Islam.   Kata Kunci: Panggilan kepada Islam, dakwah Islamiyyah, dakwah kreatif, da’ie kreatif, menyuruh kepada kebaikan, menghalang daripada kejahatan.  


Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.


2010 ◽  
Vol 59 (10) ◽  
pp. 1392-1401 ◽  
Author(s):  
Xiaofeng Liao ◽  
Fei Chen ◽  
Kwok-wo Wong

2013 ◽  
Vol 28 (10) ◽  
pp. 1537-1547 ◽  
Author(s):  
J.B. Lima ◽  
E.A.O. Lima ◽  
F. Madeiro

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