University of IsfahanInternational Journal of Group Theory2251-76507420181201Locally graded groups with a condition on infinite subsets172123410.22108/ijgt.2016.21234ENAsadollahFaramarzi SallesDamghan UniversityFatemehPazandeh ShanbehbazariDamghan UniversityJournal Article20160217Let $G$ be a group, we say that $G$ satisfies the property $\mathcal{T}(\infty)$ provided that, every infinite set of elements of $G$ contains elements $x\neq y, z$ such that $[x, y, z]=1=[y, z, x]=[z, x, y]$.<br /> We denote by $\mathcal{C}$ the class of all polycyclic groups, $\mathcal{S}$ the class of all soluble groups, $\mathcal{R}$ the class of all residually finite groups, $\mathcal{L}$ the class of all locally graded groups, $\mathcal{N}_2$ the class of all nilpotent group of class at most two, and $\mathcal{F}$ the class of all finite groups. In this paper, first we shall prove that if $G$ is a finitely generated locally graded group, then $G$ satisfies $\mathcal{T}(\infty)$ if and only if $G/Z_2(G)$ is finite, and then we shall conclude that if $G$ is a finitely generated group in $\mathcal{T}(\infty)$, then \[G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_2\mathcal{F}.\]https://ijgt.ui.ac.ir/article_21234_67c122bc31064ada379ba0fa8178aec3.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507420181201Automorphisms of a finite $p$-group with cyclic Frattini subgroup9162121910.22108/ijgt.2017.21219ENRasoulSoleimaniPayame Noor UniversityJournal Article20160808Let $G$ be a group and $Aut^{\Phi}(G)$ denote the group of all automorphisms of $G$ centralizing $G/\Phi(G)$ elementwise. In this paper, we characterize the finite $p$-groups $G$ with cyclic Frattini subgroup for which $|Aut^{\Phi}(G):Inn(G)|=p$.https://ijgt.ui.ac.ir/article_21219_ae7d67b716884474ebab05e35cda245c.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507420181201On embedding of partially commutative metabelian groups to matrix groups17262147810.22108/ijgt.2017.21478ENE. I.TimoshenkoNovosibirsk State Technical UniversityJournal Article20160306The Magnus embedding of a free metabelian group induces the embedding of partially commutative metabelian group $S_\Gamma$ in a group of matrices $M_\Gamma$. Properties and the universal theory of the group $M_\Gamma$ are studied.https://ijgt.ui.ac.ir/article_21478_06e8a271d84561be036e425c8e46cc0c.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507420181201Measuring cones and other thick subsets in free groups27402147910.22108/ijgt.2017.21479ENElizavetaFrenkelMoscow State UniversityVladimirRemeslennikovMathematical Institute SB RASJournal Article20160330In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they accept. We show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoids. These automata allow us to compute explicitly generating functions, $\lambda-$measures and Cesaro measure of thick monoids. Also we improve the asymptotic classification of regular subsets in free groups.https://ijgt.ui.ac.ir/article_21479_6002b97cd87509a69bdf9b2e53ab514f.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507420181201The Maschke property for the Sylow $p$-subgroups of the symmetric group $S_{p^n}$41642161010.22108/ijgt.2017.21610ENDavid J.GreenInstitut für Mathematik
Friedrich-SchillerüUniversität
07737 JenaL.HéthelyiBudapest University of Technology and Economics, Mathematical Institute,
Department of Algebra
H-1111 Budapest,
Műegyetem rkp. 3-9.E.HorváthBudapest University of Technology
and Economics, Faculty of Sciences,
Inst. Math., Department of Algebra,
H-1111 Budapest, Műegyetem rkp. 3-9.Journal Article20161023In this paper we prove that the Maschke property holds for coprime actions on some important classes of $p$-groups like: metacyclic $p$-groups, $p$-groups of $p$-rank two for $p>3$ and some weaker property holds in the case of regular $p$-groups. The main focus will be the case of coprime actions on the iterated wreath product $P_n$ of cyclic groups of order $p$, i.e. on Sylow $p$-subgroups of the symmetric groups $S_{p^n}$, where we also prove that a stronger form of the Maschke property holds. These results contribute to a future possible classification of all $p$-groups with the Maschke property. We apply these results to describe which normal partition subgroups of $P_n$ have a complement. In the end we also describe abelian subgroups of $P_n$ of largest size.https://ijgt.ui.ac.ir/article_21610_049d5dd2426c246d448583ee0a063476.pdf