On Huppert's conjecture for $F_4(2)$ Hung Tong-Viet North-West University, Mafikeng Campus author Thomas Wakefield Youngstown State University author text article 2012 eng Let $G$ be a finite group and let $\text{cd}(G)$ be the set of all‎ ‎complex irreducible character degrees of $G$‎. ‎B‎. ‎Huppert conjectured‎ ‎that if $H$ is a finite nonabelian simple group such that‎ ‎$\text{cd}(G) =\text{cd}(H)$‎, ‎then $G\cong H \times A$‎, ‎where $A$ is‎ ‎an abelian group‎. ‎In this paper‎, ‎we verify the conjecture for‎ ‎${F_4(2)}.$‎ International Journal of Group Theory University of Isfahan 2251-7650 1 v. 3 no. 2012 1 9 http://ijgt.ui.ac.ir/article_763_84125d4a5e2e7ac9630a1081299e34f0.pdf dx.doi.org/10.22108/ijgt.2012.763 On finite A-perfect abelian groups Mohammad Mehdi Nasrabadi Department of Maths,birjand university author Ali Gholamian Department of mathematics, Birjand university, Birjand author text article 2012 eng ‎Let $G$ be a group and $A=Aut(G)$ be the group of automorphisms of‎ ‎$G$‎. ‎Then the element $[g,\alpha]=g^{-1}\alpha(g)$ is an‎ ‎autocommutator of $g\in G$ and $\alpha\in A$‎. ‎Also‎, ‎the‎ ‎autocommutator subgroup of G is defined to be‎ ‎$K(G)=\langle[g,\alpha]|g\in G‎, ‎\alpha\in A\rangle$‎, ‎which is a‎ ‎characteristic subgroup of $G$ containing the derived subgroup‎ ‎$G'$ of $G$‎. ‎A group is defined as A-perfect‎, ‎if it equals its own‎ ‎autocommutator subgroup‎. ‎The present research is aimed at‎ ‎classifying finite abelian groups which are A-perfect‎. International Journal of Group Theory University of Isfahan 2251-7650 1 v. 3 no. 2012 11 14 http://ijgt.ui.ac.ir/article_764_c0e46ef10fc8cc8252147940e51d0023.pdf dx.doi.org/10.22108/ijgt.2012.764 A new characterization of $PSL(2, 25)$ Alireza Khalili Asboei Babol Education, Mazandaran, Iran author Syyed Sadegh Salehi Amiri Islamic Azad University Babol Branch author text article 2012 eng ‎Let $G$ be a finite group and $\pi_{e}(G)$ be the set of element‎ ‎orders of $G$‎. ‎Let $k \in \pi_{e}(G)$ and $m_{k}$ be the number of‎ ‎elements of order $k$ in $G$‎. ‎Set nse($G$):=$\{ m_{k} | k \in‎ ‎\pi_{e}(G)\}$‎. ‎In this paper‎, ‎we prove that if $G$ is a group such‎ ‎that nse($G$)=nse($PSL(2‎, ‎25)$)‎, ‎then $G \cong PSL(2‎, ‎25)$‎. International Journal of Group Theory University of Isfahan 2251-7650 1 v. 3 no. 2012 15 19 http://ijgt.ui.ac.ir/article_765_3cb589fd74c1a6fe0587c1d1dc0a64f1.pdf dx.doi.org/10.22108/ijgt.2012.765 On the semi cover-avoiding property and $\mathcal{F}$-supplementation Changwen Li xuzhou normal university author Xiaolan Yi author text article 2012 eng In this paper‎, ‎we investigate the influence of some subgroups of Sylow subgroups with semi cover-avoiding property and‎ ‎$\mathcal{F}$-supplementation on the structure of finite groups and generalize a series of known results‎. International Journal of Group Theory University of Isfahan 2251-7650 1 v. 3 no. 2012 21 31 http://ijgt.ui.ac.ir/article_996_fb53e9a8f41d8c6c5704d4c48beabbc2.pdf dx.doi.org/10.22108/ijgt.2012.996 On varietal capability of Infinite direct products of groups Hanieh Mirebrahimi Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran author Behrooz Mashayekhy Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran author text article 2012 eng Recently‎, ‎the authors gave some conditions under which a direct product‎ ‎of finitely many groups is $\mathcal{V}-$capable if and only if each of its‎ ‎factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$‎. ‎In this paper‎, ‎we extend this fact to any infinite direct product of groups‎. ‎Moreover‎, ‎we conclude some results for $\mathcal{V}-$capability of direct products of infinitely many groups in varieties of abelian‎, ‎nilpotent and polynilpotent groups‎. International Journal of Group Theory University of Isfahan 2251-7650 1 v. 3 no. 2012 33 37 http://ijgt.ui.ac.ir/article_850_c7158d00eaa639c32430a262f09c81fa.pdf dx.doi.org/10.22108/ijgt.2012.850 Infinite groups with many generalized normal subgroups Francesco de Giovanni Dipartimento di Matematica e Applicazioni - University of Napoli Federico II author Caterina Rainone Universita di Napoli Federico II author text article 2012 eng A subgroup $X$ of a group $G$ is almost normal if the index $|G:N_G(X)|$ is finite‎, ‎while $X$ is nearly normal if it has finite index in the normal closure $X^G$‎. ‎This paper investigates the structure of groups in which every (infinite) subgroup is either almost normal or nearly normal‎. International Journal of Group Theory University of Isfahan 2251-7650 1 v. 3 no. 2012 39 49 http://ijgt.ui.ac.ir/article_1223_150426f6f6fcaf626f8a811ef1dd2718.pdf dx.doi.org/10.22108/ijgt.2012.1223 Quasirecognition by prime graph of $U_3(q)$ where $2 < q =p^{\alpha} < 100$ Seyed Sadegh Salehi Amiri Islamic Azad University author Alireza Khalili Asboei Islamic Azad University author Ali Iranmanesh Tarbiat Modares University author Abolfazl Tehranian Islamic Azad University author text article 2012 eng Let $G$ be a finite group and let $\Gamma(G)$ be the prime graph‎ ‎of G‎. ‎Assume $2 < q = p^{\alpha} < 100$‎. ‎We determine finite groups‎ ‎G such that $\Gamma(G) = \Gamma(U_3(q))$ and prove that if $q \neq‎ ‎3‎, ‎5‎, ‎9‎, ‎17$‎, ‎then $U_3(q)$ is quasirecognizable by prime graph‎, ‎i.e‎. ‎if $G$ is a finite group with the same prime graph as the‎ ‎finite simple group $U_3(q)$‎, ‎then $G$ has a unique non-Abelian‎ ‎composition factor isomorphic to $U_3(q)$‎. ‎As a consequence of our‎ ‎results‎, ‎we prove that the simple groups $U_{3}(8)$ and $U_{3}(11)$‎ ‎are $4-$recognizable and $2-$recognizable by prime graph‎, ‎respectively‎. ‎In fact‎, ‎the group $U_{3}(8)$ is the first example‎ ‎which is a $4-$recognizable by prime graph‎. International Journal of Group Theory University of Isfahan 2251-7650 1 v. 3 no. 2012 51 66 http://ijgt.ui.ac.ir/article_1369_3a2c5d4f00b6ca4392b6fc4dafbcde67.pdf dx.doi.org/10.22108/ijgt.2012.1369