eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 1 12 10.22108/ijgt.2014.4570 4570 Restrictions on commutativity ratios in finite groups Robert Heffernan robert.heffernan@uconn.edu 1 Des MacHale d.machale@ucc.ie 2 Aine Ni She aine.nishe@cit.ie 3 University of Connecticut University College Cork Cork Institute of Technology  ‎We consider two commutativity ratios \$Pr(G)\$ and \$f(G)\$ in a finite group \$G\$‎ ‎and examine the properties of \$G\$ when these ratios are `large'‎. ‎We show that‎ ‎if \$Pr(G) > frac{7}{24}\$‎, ‎then \$G\$ is metabelian and we give threshold‎ ‎results in the cases where \$G\$ is insoluble and \$G'\$ is nilpotent‎. ‎We also‎ ‎show that if \$f(G) > frac{1}{2}\$‎, ‎then \$f(G) = frac{n+1}{2n}\$‎, ‎for some‎ ‎natural number \$n\$‎. http://ijgt.ui.ac.ir/article_4570_55d0f1553f00e86de09233d4129f5a8f.pdf commutativity ratios commuting probability Finite groups eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 13 16 10.22108/ijgt.2014.4776 4776 The unit group of algebra of circulant matrices Neha Makhijani nehamakhijani@gmail.com 1 R. K. 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 \$Cr_{n}(F)\$ denote the algebra of \$ntimes n\$ circulant matrices over the field \$F\$‎. ‎In this paper‎, ‎we study the unit group of \$Cr_{n}(mathbb{F}_{p^{m}})\$‎, ‎where \$mathbb{F}_{p^{m}}\$ denotes the Galois field of order \$p^{m},~p\$ prime‎. http://ijgt.ui.ac.ir/article_4776_76fd6d2530a88ac6184b7c4c0c57fca7.pdf group algebra Unit Group Circulant Matrices eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 17 25 10.22108/ijgt.2014.4950 4950 On weakly \$SS\$-quasinormal and hypercyclically embedded properties of finite groups Tao Zhao zht198109@163.com 1 School of Science, Shandong University of Technology A subgroup \$H\$ is said to be \$s\$-permutable in a group \$G\$‎, ‎if‎ ‎\$HP=PH\$ holds for every Sylow subgroup \$P\$ of \$G\$‎. ‎If there exists a‎ ‎subgroup \$B\$ of \$G\$ such that \$HB=G\$ and \$H\$ permutes with every‎ ‎Sylow subgroup of \$B\$‎, ‎then \$H\$ is said to be \$SS\$-quasinormal in‎ ‎\$G\$‎. ‎In this paper‎, ‎we say that \$H\$ is a weakly \$SS\$-quasinormal‎ ‎subgroup of \$G\$‎, ‎if there is a normal subgroup \$T\$ of \$G\$ such that‎ ‎\$HT\$ is \$s\$-permutable and \$Hcap T\$ is \$SS\$-quasinormal in \$G\$‎. ‎By‎ ‎assuming that some subgroups of \$G\$ with prime power order have the‎ ‎weakly \$SS\$-quasinormal properties‎, ‎we get some new‎ ‎characterizations about the hypercyclically embedded subgroups of‎ ‎\$G\$‎. ‎A series of known results in the literature are unified and‎ ‎generalized. http://ijgt.ui.ac.ir/article_4950_c0915a41877e3a4bb1db406fbaca42cf.pdf ‎\$s\$-permutable‎ ‎weakly \$SS\$-quasinormal‎ \$p\$-nilpotent‎ ‎hypercyclically embedded eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 27 31 10.22108/ijgt.2014.4952 4952 On zero patterns of characters of finite groups Jinshan Zhang zjscdut@163.com 1 Guangju Zeng weiwei@suse.edu.cn 2 Zhencai Shen 3 School of Science, Sichuan University of Science and Engineering, Zigong, 643000, P. R. China Sichuan University of Science and Engineering China Agricultural University The aim of this note is to characterize the finite‎ ‎groups in which all non-linear irreducible characters have distinct zero entries number‎. http://ijgt.ui.ac.ir/article_4952_63eb74c1c94bc55ca2353308a1051eba.pdf Finite groups characters zeros of characters eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 33 36 10.22108/ijgt.2014.4976 4976 A note on the normalizer of Sylow \$2\$-subgroup of special linear‎ ‎group \${rm SL}_2(p^f)\$ Jiangtao Shi jiangtaoshi@126.com 1 Yantai University Let \$G={rm SL}_2(p^f)\$ be a special linear group and \$P\$ be a Sylow‎ ‎\$2\$-subgroup of \$G\$‎, ‎where \$p\$ is a prime and \$f\$ is a positive‎ ‎integer such that \$p^f>3\$‎. ‎By \$N_G(P)\$ we denote the normalizer of‎ ‎\$P\$ in \$G\$‎. ‎In this paper‎, ‎we show that \$N_G(P)\$ is nilpotent (or‎ ‎\$2\$-nilpotent‎, ‎or supersolvable) if and only if \$p^{2f}equiv‎ ‎1,({rm mod},16)\$‎. http://ijgt.ui.ac.ir/article_4976_a69c9b523546d6cc0812f1d9027240e7.pdf special linear group Sylow subgroup normalizer nilpotent supersolvable eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 37 46 10.22108/ijgt.2014.5087 5087 On one class of modules over group rings with finiteness restrictions Olga Dashkova odashkova@yandex.ru 1 Professor of the Branch of Moscow state university in Sevastopol The author studies the \$bf R\$\$G\$-module \$A\$ such that \$bf R\$ is an associative ring‎, ‎a group \$G\$ has infinite section \$p\$-rank (or infinite 0-rank)‎, ‎\$C_{G}(A)=1\$‎, ‎and for every‎ ‎proper subgroup \$H\$ of infinite section \$p\$-rank (or infinite 0-rank respectively) the quotient module \$A/C_{A}(H)\$ is‎ ‎a finite \$bf R\$-module‎. ‎It is proved that if the group \$G\$ under‎ ‎consideration is locally soluble‎ ‎then \$G\$ is a soluble group and \$A/C_{A}(G)\$ is a finite \$bf R\$-module‎. ‎ http://ijgt.ui.ac.ir/article_5087_ca4189aa5efbaeed67562c6122922f8a.pdf group ring linear group module eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 47 56 10.22108/ijgt.2014.5254 5254 Quasirecognition by prime graph of finite simple groups \${}^2D_n(3)\$ Behrooz Khosravi bkhosravi@aut.ac.ir 1 Hossein Moradi khosravibbb@yahoo.com 2 Amirkabir University of Technology ‎Let \$G\$ be a finite group‎. ‎In [Ghasemabadi et al.‎, ‎characterizations of the simple group \${}^2D_n(3)\$ by prime graph‎ ‎and spectrum‎, ‎Monatsh Math.‎, ‎2011] it is‎ ‎proved that if \$n\$ is odd‎, ‎then \${}^2D _n(3)\$ is recognizable by‎ ‎prime graph and also by element orders‎. ‎In this paper we prove‎ ‎that if \$n\$ is even‎, ‎then \$D={}^2D_{n}(3)\$ is quasirecognizable by‎ ‎prime graph‎, ‎i.e‎. ‎every finite group \$G\$ with \$Gamma(G)=Gamma(D)\$‎ ‎has a unique nonabelian composition factor and this factor is isomorphic to‎ ‎\$D\$‎. http://ijgt.ui.ac.ir/article_5254_b31e2bb7e4d6f7188c9fd129dd78758f.pdf Prime graph simple group linear group quasirecognition eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 57 61 10.22108/ijgt.2014.5342 5342 A note on fixed points of automorphisms of infinite groups Francesco de Giovanni degiovan@unina.it 1 Martin L. Newell martin.newell@nuigalway.ie 2 Alessio Russo alessio.russo@unina2.it 3 University of Napoli Federico II National University of Ireland Seconda Universita di Napoli ‎Motivated by a celebrated theorem of Schur‎, ‎we show that if \$Gamma\$ is a normal subgroup of the full automorphism group \$Aut(G)\$ of a group \$G\$ such that \$Inn(G)\$ is contained in \$Gamma\$ and \$Aut(G)/Gamma\$ has no uncountable abelian subgroups of prime exponent‎, ‎then \$[G,Gamma]\$ is finite‎, ‎provided that the subgroup consisting of all elements of \$G\$ fixed by \$Gamma\$ has finite index‎. ‎Some applications of this result are also given.‎ http://ijgt.ui.ac.ir/article_5342_1e6c5c18b97f38824f43a2febfd71900.pdf automorphism group Schur's theorem absolute centre eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2014-12-01 3 4 63 69 10.22108/ijgt.2014.5479 5479 Symmetry classes of polynomials associated with the ‎direct ‎product of permutation groups Esmaeil Babaei e_babaei@sut.ac.ir 1 Yousef Zamani zamani@sut.ac.ir 2 Sahand University of technology Sahand University of Technology ‎Let \$G_{i} \$ be a subgroup of \$ S_{m_{i}}‎ ,‎ 1 leq i leq k\$‎. ‎Suppose \$chi_{i}\$ is an irreducible complex character of \$G_{i}\$‎. ‎We consider \$ G_{1}times cdots times G_{k} \$ as subgroup of \$ S_{m} \$‎, ‎where \$ m=m_{1}+cdots‎ +‎m_{k} \$‎. ‎In this paper‎, ‎we give a formula for the dimension of \$H_{d}(G_{1}times cdots times G_{k}‎, ‎chi_{1}timescdots times chi_{k})\$ and investigate the existence of an o-basis of this type of classes‎. http://ijgt.ui.ac.ir/article_5479_0495fb15f251988634840c9c7812f01e.pdf Symmetric polynomials symmetry class of polynomials‎ ‎orthogonal basis ‎permutaion groups‎ ‎complex characters