Unit group of algebra of circulant matrices Rajendra Sharma Indian Institute of Technology Delhi author Pooja Yadav Department of Mathematics, Kamla Nehru College, University of Delhi, Delhi author text article 2013 eng Let $Cr_n(F_p)$ 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. International Journal of Group Theory University of Isfahan 2251-7650 2 v. 4 no. 2013 1 6 http://ijgt.ui.ac.ir/article_2643_33e935a9ca272310a728fc6513a0bbad.pdf dx.doi.org/10.22108/ijgt.2013.2643 Partially $S$-embedded minimal subgroups of finite groups Tao Zhao School of Science, Shandong University of Technology author Qingliang Zhang School of Sciences, Nantong University author text article 2013 eng Suppose 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.‎ International Journal of Group Theory University of Isfahan 2251-7650 2 v. 4 no. 2013 7 16 http://ijgt.ui.ac.ir/article_2751_21631b0fa51b75065747f61c434fd5e4.pdf dx.doi.org/10.22108/ijgt.2013.2751 Noninner automorphisms of finite $p$-groups leaving the center elementwise fixed Alireza Abdollahi University of Isfahan author S. Mohsen Ghoraishi University of Isfahan author text article 2013 eng A 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. International Journal of Group Theory University of Isfahan 2251-7650 2 v. 4 no. 2013 17 20 http://ijgt.ui.ac.ir/article_2761_52bac5d4d3e407efd00cc7724a0d360e.pdf dx.doi.org/10.22108/ijgt.2013.2761 On supersolvability of finite groups with $\Bbb P$-subnormal subgroups Viktoryia Kniahina Gomel engineering institute of MES of Republic of Belarus author Victor Monakhov Department of Mathematics, Gomel F. Scorina State University author text article 2013 eng In this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $\Bbb P$-subnormality guarantees supersolvability‎ ‎of the whole group‎. International Journal of Group Theory University of Isfahan 2251-7650 2 v. 4 no. 2013 21 29 http://ijgt.ui.ac.ir/article_2835_846acc825bf3d7fa7d1fe37251836e69.pdf dx.doi.org/10.22108/ijgt.2013.2835 On the probability of being a $2$-Engel group Ahmad Erfanian Ferdowsi University of Mashhad author Mohammad Farrokhi Derakhshandeh Ghouchan Ferdowsi University of Mashhad author text article 2013 eng ‎Let $G$ 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. International Journal of Group Theory University of Isfahan 2251-7650 2 v. 4 no. 2013 31 38 http://ijgt.ui.ac.ir/article_2836_e178af16ad25afc5f74265a501ad63fb.pdf dx.doi.org/10.22108/ijgt.2013.2836 On finite C-tidy groups Sekhar Baishya North-Eastern Hill University author text article 2013 eng A 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. International Journal of Group Theory University of Isfahan 2251-7650 2 v. 4 no. 2013 39 41 http://ijgt.ui.ac.ir/article_2838_8f2b0e559e4fdea04fd0b7d3c5134624.pdf dx.doi.org/10.22108/ijgt.2013.2838 The $n$-ary adding machine and solvable groups Josimar da Silva Rocha Instituto Federal de Educacao author Said Sidki Universidade De Brasilia author text article 2013 eng We 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. International Journal of Group Theory University of Isfahan 2251-7650 2 v. 4 no. 2013 43 88 http://ijgt.ui.ac.ir/article_2871_635f8d519f354a9c3204c92c000157ee.pdf dx.doi.org/10.22108/ijgt.2013.2871