Finite groups whose minimal subgroups are weakly $\mathcal{H}^{\ast}$-subgroups Abdelrahman Heliel Department of Mathematics, Faculty of Science, Beni-Suef university author Rola Hijazi Department of Mathematics, Faculty of Science, KAU, Saudi Arabia author Reem Al-Obidy Department of Mathematics, Faculty of Science, KAU, Saudi Arabia author text article 2014 eng Let $G$ be a finite group‎. ‎A subgroup‎ ‎$H$ of $G$ is called an $\mathcal{H}$-subgroup in‎ ‎$G$ if $N_G(H)\cap H^{g}\leq H$ for all $g\in‎ ‎G$. A subgroup $H$ of $G$ is called a weakly‎ ‎$\mathcal{H}^{\ast}$-subgroup in $G$ if there exists a‎ ‎subgroup $K$ of $G$ such that $G=HK$ and $H\cap‎ ‎K$ is an $\mathcal{H}$-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 ‎$\mathcal{H}^{\ast}$-subgroup in $G$. Our results improve‎ ‎and extend a series of recent results in the literature‎. International Journal of Group Theory University of Isfahan 2251-7650 3 v. 3 no. 2014 1 11 http://ijgt.ui.ac.ir/article_3837_4ba7139afccee4a6543ffa5a60f76f6d.pdf dx.doi.org/10.22108/ijgt.2014.3837 The coprime graph of a group Xuan Long Ma Beijing Normal University author Hua Quan Wei Guangxi University author Li Ying Yang Guangxi Teachers Education University author text article 2014 eng The coprime graph $\gg$ with a finite group $G$‎ ‎as follows‎: ‎Take $G$ as the vertex set of $\gg$ and join two distinct‎ ‎vertices $u$ and $v$ if $(|u|,|v|)=1$‎. ‎In the paper‎, ‎we explore how the graph‎ ‎theoretical properties of $\gg$ can effect on the group theoretical‎ ‎properties of $G$‎. International Journal of Group Theory University of Isfahan 2251-7650 3 v. 3 no. 2014 13 23 http://ijgt.ui.ac.ir/article_4363_c9f7b91082201904334198ebeaf4569a.pdf dx.doi.org/10.22108/ijgt.2014.4363 Units in $F_{2^k}D_{2n}$ Neha Makhijani Indian Institute of Technology Delhi Hauz Khas, New Delhi-110016 India author R. Sharma Indian Institute of Technology Delhi Hauz Khas, New Delhi India author J. B. Srivastava Indian Institute of Technology Delhi Hauz Khas, New Delhi India author text article 2014 eng Let $\mathbb{F}_{q}D_{2n}$ be the group algebra of $D_{2n}$‎, ‎the dihedral group of order $2n$‎, ‎over $\mathbb{F}_{q}=GF(q)$‎. ‎In this paper‎, ‎we establish the structure of $\mathcal{U}(\mathbb{F}_{2^{k}}D_{2n})$‎, ‎the unit group of $\mathbb{F}_{2^{k}}D_{2n}$ and that of its normalized unitary subgroup $V_{*}(\mathbb{F}_{2^{k}}D_{2n})$ with respect to canonical involution $*$ when $n$ is odd‎. International Journal of Group Theory University of Isfahan 2251-7650 3 v. 3 no. 2014 25 34 http://ijgt.ui.ac.ir/article_4382_381809516a5a923153e331a4094a2b06.pdf dx.doi.org/10.22108/ijgt.2014.4382 On $n$-Kappe groups Asadollah Faramarzi Salles Damghan University author Hassan Khosravi Gonbad-e Qabus University author text article 2014 eng Let $G$ be an infinite group and $n\in \{3‎, ‎6\}\cup\{2^k| k\in \mathbb{N}\}$‎. ‎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 $x\in X$ and $y\in Y$ such that $[x^n‎, ‎y‎, ‎y]=1$‎. International Journal of Group Theory University of Isfahan 2251-7650 3 v. 3 no. 2014 35 38 http://ijgt.ui.ac.ir/article_4434_9a17cff5f39ae447a1760a128a838980.pdf dx.doi.org/10.22108/ijgt.2014.4434 Splitting of extensions in the category of locally compact abelian groups Hossein Sahleh Department of Mathematics University of Guilan author Akbar Alijani University of Guilan author text article 2014 eng 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 $0\to X\to Y\to G\to 0$ with divisible‎, ‎$\sigma-$compact $X$ in $\pounds$ splits‎. ‎We also determine the discrete or compactly generated LCA groups $H$ such that every pure extension $0\to H\to Y\to X\to 0$ splits for each divisible group $X$ in $\pounds$‎. International Journal of Group Theory University of Isfahan 2251-7650 3 v. 3 no. 2014 39 45 http://ijgt.ui.ac.ir/article_4435_22e1c3d0e09071cafadb580f175adec2.pdf dx.doi.org/10.22108/ijgt.2014.4435 On the total character of finite groups Sunil Prajapati NBHM Postdoctoral fellow in Indian Statistical Institute Bangalore (I have submitted my PhD thesis at Indian Institute of Technology Delhi). author Balasubramanian Sury Indian Statistical Institute bangalore, India author text article 2014 eng 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. International Journal of Group Theory University of Isfahan 2251-7650 3 v. 3 no. 2014 47 67 http://ijgt.ui.ac.ir/article_4446_e0d321ff268fc949ff98a187267f48e3.pdf dx.doi.org/10.22108/ijgt.2014.4446