University of IsfahanInternational Journal of Group Theory2251-76502420131201Unit group of algebra of circulant matrices16264310.22108/ijgt.2013.2643ENRajendraSharmaIndian Institute of Technology DelhiPoojaYadavDepartment of Mathematics,
Kamla Nehru College,
University of Delhi, DelhiJournal Article20120928Let $Cr_n(F_p)$ denote the algebra of $n \times n$ circulant matrices over $F_p$, the finite field of order $p$ a prime. The order of the unit groups $\mathcal{U}(Cr_3(F_p))$, $\mathcal{U}(Cr_4(F_p))$ and $\mathcal{U}(Cr_5(F_p))$ of algebras of circulant matrices over $F_p$ are computed.https://ijgt.ui.ac.ir/article_2643_33e935a9ca272310a728fc6513a0bbad.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76502420131201Partially $S$-embedded minimal subgroups of finite groups716275110.22108/ijgt.2013.2751ENTaoZhaoSchool of Science, Shandong University of TechnologyQingliangZhangSchool of Sciences, Nantong UniversityJournal Article20130213Suppose that $H$ is a subgroup of $G$, then $H$ is said to be $s$-permutable in $G$, if $H$ permutes with every Sylow subgroup of $G$. If $HP=PH$ hold for every Sylow subgroup $P$ of $G$ with $(|P|, |H|)=1$), then $H$ is called an $s$-semipermutable subgroup of $G$. In this paper, we say that $H$ is partially $S$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $HT$ is $s$-permutable in $G$ and $H\cap T\leq H_{\overline{s}G}$, where $H_{\overline{s}G}$ is generated by all $s$-semipermutable subgroups of $G$ contained in $H$. We investigate the influence of some partially $S$-embedded minimal subgroups on the nilpotency and supersolubility of a finite group $G$. A series of known results in the literature are unified and generalized.https://ijgt.ui.ac.ir/article_2751_21631b0fa51b75065747f61c434fd5e4.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76502420131201Noninner automorphisms of finite p-groups leaving the center elementwise fixed1720276110.22108/ijgt.2013.2761ENAlirezaAbdollahiUniversity of IsfahanS. MohsenGhoraishiUniversity of IsfahanJournal Article20130225A longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$. Let $G$ be a finite nonabelian $p$-group. It is known that if $G$ is regular or of nilpotency class $2$ or the commutator subgroup of $G$ is cyclic, or $G/Z(G)$ is powerful, then $G$ has a noninner automorphism of order $p$ leaving either the center $Z(G)$ or the Frattini subgroup $\Phi(G)$ of $G$ elementwise fixed. In this note, we prove that the latter noninner automorphism can be chosen so that it leaves $Z(G)$ elementwise fixed.https://ijgt.ui.ac.ir/article_2761_52bac5d4d3e407efd00cc7724a0d360e.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76502420131201On supersolvability of finite groups with ℙ-subnormal subgroups2129283510.22108/ijgt.2013.2835ENViktoryiaKniahinaGomel engineering institute of MES of Republic of BelarusVictorMonakhovDepartment of
Mathematics, Gomel F. Scorina State UniversityJournal Article20121201In this paper we find systems of subgroups of a finite group, which $\Bbb P$-subnormality guarantees supersolvability of the whole group.https://ijgt.ui.ac.ir/article_2835_846acc825bf3d7fa7d1fe37251836e69.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76502420131201On the probability of being a 2-Engel group3138283610.22108/ijgt.2013.2836ENAhmadErfanianFerdowsi University of MashhadMohammadFarrokhi Derakhshandeh GhouchanFerdowsi University of MashhadJournal Article20130312Let $G$ be a finite group and $d_2(G)$ denotes the probability that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $G$. We will obtain lower and upper bounds for $d_2(G)$ in the case where the sets $E_G(x)=\{y\in G:[y,x,x]=1\}$ are subgroups of $G$ for all $x\in G$. Also the given examples illustrate that all the bounds are sharp.https://ijgt.ui.ac.ir/article_2836_e178af16ad25afc5f74265a501ad63fb.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76502420131201On finite C-tidy groups3941283810.22108/ijgt.2013.2838ENSekhar JyotiBaishyaNorth-Eastern Hill UniversityJournal Article20130130A group $G$ is said to be a C-tidy group if for every element $x \in G \setminus K(G)$, the set $Cyc(x)=\lbrace y \in G \mid \langle x, y \rangle \; {\rm is \; cyclic} \rbrace$ is a cyclic subgroup of $G$, where $K(G)=\underset{x \in G}\bigcap Cyc(x)$. In this short note we determine the structure of finite C-tidy groups.https://ijgt.ui.ac.ir/article_2838_8f2b0e559e4fdea04fd0b7d3c5134624.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76502420131201The $n$-ary adding machine and solvable groups4388287110.22108/ijgt.2013.2871ENJosimarDa Silva RochaInstituto Federal de EducacaoSaidSidkiUniversidade De BrasiliaJournal Article20130420We describe under various conditions abelian subgroups of the automorphism group $\mathrm{Aut}(T_{n})$ of the regular $n$-ary tree $T_{n}$, which are normalized by the $n$-ary adding machine $\tau =(e, \dots, e,\tau )\sigma _{\tau }$ where $\sigma _{\tau }$ is the $n$-cycle $\left( 0,1, \dots, n-1\right) $. As an application, for $n=p$ a prime number, and for $n=4$, we prove that every soluble subgroup of $\mathrm{Aut}(T_{n})$, containing $\tau $ is an extension of a torsion-free metabelian group by a finite group.https://ijgt.ui.ac.ir/article_2871_635f8d519f354a9c3204c92c000157ee.pdf