University of IsfahanInternational Journal of Group Theory2251-76508120190113On noninner automorphisms of finite $p$-groups that fix the center elementwise192241210.22108/ijgt.2018.108082.1457ENS. MohsenGhoraishiShahid Chamran UniversityJournal Article20171119In this paper we show that every finite nonabelian $p$-group $G$ in which the Frattini subgroup $\Phi(G)$ has order $\leq p^5$ admits a noninner automorphism of order $p$ leaving the center $Z(G)$ elementwise fixed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least $p^8$.https://ijgt.ui.ac.ir/article_22412_bb6fd44e271320636320a62cf00c7ba2.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508120190301On the ranks of Fischer group $Fi_{24}^{\,\prime}$ and the Baby Monster group $\mathbb{B}$11222270910.22108/ijgt.2018.109973.1471ENMohammed Ali FayaIbrahimNajran UniversityFaryadAliAl Imam Mohammad Ibn Saud Islamic UniversityMohammed A.Al-KadhiAl Imam Mohammad Ibn Saud Islamic UniversityAbdullah MohammedAljouieeAl Imam Mohammed Ibn Saud Islamic UniversityJournal Article20180301If $G$ is a finite group and $X$ a conjugacy class of elements of $G$, then we define $rank(G{:}X)$ to be the minimum number of elements of $X$ generating $G$. In the present article, we determine the ranks for the Fischer's simple group $Fi_{24}^{\prime}$ and the baby monster group $\mathbb{B}$.https://ijgt.ui.ac.ir/article_22709_90a14c8661d74f129eb1fe4ba24834bb.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508120190301Finite groups with the same conjugacy class sizes as a finite simple group23332123610.22108/ijgt.2017.21236ENNedaAhanjidehUniversity of Shahre-kordJournal Article20160501For a finite group $H$, let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$. In this paper, we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$, then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$. In particular, we show that for some finite simple group $S$, $G \cong S \times Z(G)$.https://ijgt.ui.ac.ir/article_21236_1a260ed6b265defd6362bf30cf33c9ce.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508120190301Finite groups of the same type as Suzuki groups35422155610.22108/ijgt.2017.21556ENSeyed HassanAlaviDepartment of Mathematics, Bu-Ali Sina University, Hamedan, IranAshrafDaneshkhahDepartment of Mathematics, Bu-Ali Sina University, Hamedan, IranHoseinParvizi MosaedAlvand Institute of Higher Education, Hamedan, Iran.Journal Article20170308For a finite group $G$ and a positive integer $n$, let $G(n)$ be the set of all elements in $G$ such that $x^{n}=1$. The groups $G$ and $H$ are said to be of the same (order) type if $|G(n)|=|H(n)|$, for all $n$. The main aim of this paper is to show that if $G$ is a finite group of the same type as Suzuki groups $Sz(q)$, where $q=2^{2m+1}\geq 8$, then $G$ is isomorphic to $Sz(q)$. This addresses to the well-known J. G. Thompson's problem (1987) for simple groups.https://ijgt.ui.ac.ir/article_21556_63d2193bfc7047cda3611ae9155ce682.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508120190301Difference bases in dihedral groups43502161210.22108/ijgt.2017.21612ENTaras O.BanakhIvan Franko National University of Lviv (Ukraine),
and Institute of Mathematics, Jan Kochanowski University in
Kielce (Poland)VolodymyrGavrylkivVasyl Stefanyk Precarpathian National
University, Ivano-Frankivsk, UkraineJournal Article20170412A subset $B$ of a group $G$ is called a {\em difference basis} of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the {\em difference size} of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\Delta[G]/{\sqrt{|G|}}$ is called the {\em difference characteristic} of $G$. We prove that for every $n\in N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $\sqrt{2}\le\eth[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx1.983$. Moreover, if $n\ge 2\cdot 10^{15}$, then $\eth[D_{2n}]<\frac{4}{\sqrt{6}}\approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le80$.https://ijgt.ui.ac.ir/article_21612_09abda43cc316d9fccf8681b5cc2872d.pdf