eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2018-12-01 7 4 1 7 10.22108/ijgt.2016.21234 21234 Locally graded groups with a condition on infinite subsets Asadollah Faramarzi Salles faramarzi@du.ac.ir 1 Fatemeh Pazandeh Shanbehbazari fateme.pazandeh@gmail.com 2 Damghan University Damghan University Let \$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 \$xneq y‎, ‎z\$ such that \$[x‎, ‎y‎, ‎z]=1=[y‎, ‎z‎, ‎x]=[z‎, ‎x‎, ‎y]\$‎. ‎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 ‎[Ginmathcal{L}Leftrightarrow Ginmathcal{R}Leftrightarrow Ginmathcal{S}Leftrightarrow Ginmathcal{C}Leftrightarrow Ginmathcal{N}_2mathcal{F}.]‎ http://ijgt.ui.ac.ir/article_21234_67c122bc31064ada379ba0fa8178aec3.pdf ‎Finitely generated groups‎ ‎Residually finite groups‎ ‎Locally graded groups eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2018-12-01 7 4 9 16 10.22108/ijgt.2017.21219 21219 Automorphisms of a finite \$p\$-group with cyclic Frattini subgroup Rasoul Soleimani rsoleimanii@yahoo.com 1 Payame Noor University Let \$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\$‎. http://ijgt.ui.ac.ir/article_21219_ae7d67b716884474ebab05e35cda245c.pdf ‎‎Automorphism group‎ ‎Finite \$p\$-group‎ ‎Frattini subgroup‎ eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2018-12-01 7 4 17 26 10.22108/ijgt.2017.21478 21478 On embedding of partially commutative metabelian groups to matrix groups E. I. Timoshenko eitim45@gmail.com 1 Novosibirsk State Technical University ‎The 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. http://ijgt.ui.ac.ir/article_21478_06e8a271d84561be036e425c8e46cc0c.pdf Partially commutative group Metabeliah group universal theory Equations in group eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2018-12-01 7 4 27 40 10.22108/ijgt.2017.21479 21479 Measuring cones and other thick subsets in free groups Elizaveta Frenkel lizzy.frenkel@gmail.com 1 Vladimir Remeslennikov remesl@ofim.oscsbras.ru 2 Moscow State University Mathematical Institute SB RAS In 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. http://ijgt.ui.ac.ir/article_21479_6002b97cd87509a69bdf9b2e53ab514f.pdf free group ‎\$lambda-\$measure regular subset special automaton thick monoid eng University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 2018-12-01 7 4 41 64 10.22108/ijgt.2017.21610 21610 The Maschke property for the Sylow \$p\$-sub-groups of the symmetric group \$S_{p^n}\$ David Green david.green@uni-jena.de 1 ‎L. H&#039;ethelyi fobaba@t-online.hu 2 E. Horv&#039;ath he@math.bme.hu 3 Institut f&uuml;r Mathematik Friedrich-Schiller&uuml;Universit&auml;t 07737 Jena Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9. Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., Department of Algebra, H-1111 Budapest, Műegyetem rkp. 3-9. ‎‎In 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. http://ijgt.ui.ac.ir/article_21610_049d5dd2426c246d448583ee0a063476.pdf ‎Maschke's Theorem‎ ‎coprime action‎ ‎Sylow \$p\$-subgroup of symmetric group‎ ‎iterated wreath product‎ ‎uniserial action