eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-09-01 3 3 1 11 10.22108/ijgt.2014.3837 3837 Finite groups whose minimal subgroups are weakly \$mathcal{H}^{ast}\$-subgroups Abdelrahman Heliel heliel9@yahoo.com 1 Rola Hijazi rhijazi@kau.edu.sa 2 Reem Al-Obidy r.alobidy.1988@hotmail.com 3 Department of Mathematics, Faculty of Science, Beni-Suef university Department of Mathematics, Faculty of Science, KAU, Saudi Arabia Department of Mathematics, Faculty of Science, KAU, Saudi Arabia 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 \$gin‎ ‎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 \$Hcap‎ ‎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. http://ijgt.ui.ac.ir/article_3837_4ba7139afccee4a6543ffa5a60f76f6d.pdf ‎weakly \$mathcal{H}\$-subgroup weakly‎ ‎\$mathcal{H}^{ast}\$-subgroup‎ ‎c-supplemented subgroup‎ ‎generalized‎ ‎Fitting subgroup‎ ‎saturated formation eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-09-01 3 3 13 23 10.22108/ijgt.2014.4363 4363 The coprime graph of a group Xuan Long Ma mxl881112@126.com 1 Hua Quan Wei weihuaquan@163.com 2 Li Ying Yang yangliying@163.com 3 Beijing Normal University Guangxi University Guangxi Teachers Education University 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\$‎. http://ijgt.ui.ac.ir/article_4363_c9f7b91082201904334198ebeaf4569a.pdf coprime graph Finite group automorphism group eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-09-01 3 3 25 34 10.22108/ijgt.2014.4382 4382 Units in \$F_{2^k}D_{2n}\$ Neha Makhijani nehamakhijani@gmail.com 1 R. Sharma rksharmaiitd@gmail.com 2 J. B. Srivastava jbsrivas@gmail.com 3 Indian Institute of Technology Delhi Hauz Khas, New Delhi-110016 India Indian Institute of Technology Delhi Hauz Khas, New Delhi India Indian Institute of Technology Delhi Hauz Khas, New Delhi India 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. http://ijgt.ui.ac.ir/article_4382_381809516a5a923153e331a4094a2b06.pdf group algebra Unit Group Unitary Units eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-09-01 3 3 35 38 10.22108/ijgt.2014.4434 4434 On \$n\$-Kappe groups Asadollah Faramarzi Salles faramarzi@du.ac.ir 1 Hassan Khosravi hassan_khosravy@yahoo.com 2 Damghan University Gonbad-e Qabus University 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\$‎. http://ijgt.ui.ac.ir/article_4434_9a17cff5f39ae447a1760a128a838980.pdf ‎Kappe groups‎ ‎Variety of groups‎ ‎Erd"{o}s's Problem eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-09-01 3 3 39 45 10.22108/ijgt.2014.4435 4435 Splitting of extensions in the category of locally compact abelian groups Hossein Sahleh sahleh@guilan.ac.ir 1 Akbar Alijani taleshalijan@phd.guilan.ac.ir 2 Department of Mathematics University of Guilan University of Guilan 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\$‎. http://ijgt.ui.ac.ir/article_4435_22e1c3d0e09071cafadb580f175adec2.pdf Locally compact abelian group Splitting of Extension Divisible group eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-09-01 3 3 47 67 10.22108/ijgt.2014.4446 4446 On the total character of finite groups Sunil Prajapati skprajapati.iitd@gmail.com 1 Balasubramanian Sury sury@isibang.ac.in 2 NBHM Postdoctoral fellow in Indian Statistical Institute Bangalore (I have submitted my PhD thesis at Indian Institute of Technology Delhi). Indian Statistical Institute bangalore, India 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. http://ijgt.ui.ac.ir/article_4446_e0d321ff268fc949ff98a187267f48e3.pdf Finite groups Group Characters Total Characters