فهرست مطالب

  • Volume:3 Issue:3, 2014
  • تاریخ انتشار: 1392/12/10
  • تعداد عناوین: 6
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  • Abdelrahman Abdelhamid Heliel, Rola Asaad Hijazi, Reem Abdulaziz Al-Obidy Pages 1-11
    Let G be a finite group. A subgroup H of G is called an H-subgroup in G if N_{G}(H)∩H^{g}≤H for all g∈G. A subgroup H of G is called a weakly H^{∗}-subgroup in G if there exists a subgroup K of G such that G=HK and H∩K is an H-subgroup in G. We investigate the structure of the finite group G under the assumption that every cyclic subgroup of G of prime order p or of order 4 (if p=2) is a weakly H^{∗}-subgroup in G. Our results improve and extend a series of recent results in the literature.
    Keywords: weakly H, subgroup, weakly H^{∗}, subgroup, c, supplemented subgroup, generalized Fitting subgroup, saturated formation
  • Xuan Long Ma, Hua Quan Wei, Li Ying Yang Pages 13-23
    The coprime graph $gg$ with a finite group $G$ as follows: Take $G$ as the vertices of $gg$ and join two distinct vertices $u$ and $v$ if $(|u|,|v|)=1$. In this paper, we want to explore how the graph theoretical properties of $gg$ can effect on the group theoretical properties of $G$. First, we investigate some properties of $gg$ and characterize certain finite groups whose the coprime graph have some property. Then, we discuss some interesting questions with the automorphism groups of coprime graphs. Finally, we prove some general graph theoretical properties of the coprime graphs of the dihedral groups.
    Keywords: Coprime graph, Finite group, Automorphism group
  • Neha Makhijani, R. Sharma, J. B. Srivastava Pages 25-34
    Let $F_q D_{2n}$ be the group algebra of $D_{2n}$, the dihedral group of order $2n$ over $F_q=GF(q)$. In this paper, we establish the structure of $U(F_{2^k}D_{2n})$, the unit group of $F_{2^k}D_{2n}$ and that of its normalized unitary subgroup $V_*(F_{2^k}D_{2n})$ with respect to canonical involution $*$ when $n$ is odd.
    Keywords: Group Algebra, Unit Group, Unitary Units
  • Asadollah Faramarzi Salles, Hassan Khosravi Pages 35-38
    Let $G$ be an infinite group and $nin {3, 6}cup{2^k| kin mathbb{N}}$. In this paper, we prove that $G$ is an $n$-Kappe group if and only if for any two infinite subsets $X$ and $Y$ of $G$, there exist $xin X$ and $yin Y$ such that $[x^n, y, y]=1$.
    Keywords: ýKappe groupsý, ýVariety of groupsý, ýErd{o}s's Problem
  • Hossein Sahleh, Akbar Alijani Pages 39-45
    Let $pounds$ be the category of all locally compact abelian (LCA) groups. In this paper, the groups $G$ in $pounds$ are determined such that every extension $0to Xto Yto Gto 0$ with divisible, $sigma-$compact $X$ in $pounds$ splits. We also determine the discrete or compactly generated LCA groups $H$ such that every pure extension $0to Hto Yto Xto 0$ splits for each divisible group $X$ in $pounds$.
    Keywords: Locally compact abelian group, Splitting of Extension, Divisible group
  • Sunil Kumar Prajapati, Balasubramanian Sury Pages 47-67
    For a finite group $G$, we study the total character $tau_G$ afforded by the direct sum of all the non-isomorphic irreducible complex representations of $G$. We resolve for several classes of groups (the Camina $p$-groups, the generalized Camina $p$-groups, the groups which admit $(G,Z(G))$ as a generalized Camina pair), the problem of existence of a polynomial $f(x) in mathbb{Q}[x]$ such that $f(chi) = tau_G$ for some irreducible character $chi$ of $G$. As a consequence, we completely determine the $p$-groups of order at most $p^5$ (with $p$ odd) which admit such a polynomial. We deduce the characterization that these are the groups $G$ for which $Z(G)$ is cyclic and $(G,Z(G))$ is a generalized Camina pair and, we conjecture that this holds good for $p$-groups of any order.
    Keywords: Finite groups, Group Characters, Total Characters