University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 4 2018 12 01 Locally graded groups with a condition on infinite subsets 1 7 EN Asadollah Faramarzi Salles Damghan University faramarzi@du.ac.ir Fatemeh Pazandeh Shanbehbazari Damghan University fateme.pazandeh@gmail.com 10.22108/ijgt.2016.21234 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}.]‎ ‎Finitely generated groups‎,‎Residually finite groups‎,‎Locally graded groups http://ijgt.ui.ac.ir/article_21234.html http://ijgt.ui.ac.ir/article_21234_67c122bc31064ada379ba0fa8178aec3.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 4 2018 12 01 Automorphisms of a finite \$p\$-group with cyclic Frattini subgroup 9 16 EN Rasoul Soleimani Payame Noor University rsoleimanii@yahoo.com 10.22108/ijgt.2017.21219 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\$‎. ‎‎Automorphism group‎,‎Finite \$p\$-group‎,‎Frattini subgroup‎ http://ijgt.ui.ac.ir/article_21219.html http://ijgt.ui.ac.ir/article_21219_ae7d67b716884474ebab05e35cda245c.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 4 2018 12 01 On embedding of partially commutative metabelian groups to matrix groups 17 26 EN E. I. Timoshenko Novosibirsk State Technical University eitim45@gmail.com 10.22108/ijgt.2017.21478 ‎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. Partially commutative group,Metabeliah group,universal theory,Equations in group http://ijgt.ui.ac.ir/article_21478.html http://ijgt.ui.ac.ir/article_21478_06e8a271d84561be036e425c8e46cc0c.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 4 2018 12 01 Measuring cones and other thick subsets in free groups 27 40 EN Elizaveta Frenkel Moscow State University lizzy.frenkel@gmail.com Vladimir Remeslennikov Mathematical Institute SB RAS remesl@ofim.oscsbras.ru 10.22108/ijgt.2017.21479 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. free group,‎\$lambda-\$measure,regular subset,special automaton,thick monoid http://ijgt.ui.ac.ir/article_21479.html http://ijgt.ui.ac.ir/article_21479_6002b97cd87509a69bdf9b2e53ab514f.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 4 2018 12 01 The Maschke property for the Sylow \$p\$-sub-groups of the symmetric group \$S_{p^n}\$ 41 64 EN David J. Green Institut f&uuml;r Mathematik Friedrich-Schiller&uuml;Universit&auml;t 07737 Jena david.green@uni-jena.de ‎L. H'ethelyi Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9. fobaba@t-online.hu E. Horv'ath Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., Department of Algebra, H-1111 Budapest, Műegyetem rkp. 3-9. he@math.bme.hu 10.22108/ijgt.2017.21610 ‎‎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. ‎Maschke's Theorem‎,‎coprime action‎,‎Sylow \$p\$-subgroup of symmetric group‎,‎iterated wreath product‎,‎uniserial action http://ijgt.ui.ac.ir/article_21610.html http://ijgt.ui.ac.ir/article_21610_049d5dd2426c246d448583ee0a063476.pdf