Int. J. Group Theory University of Isfahan International Journal of Group Theory 2251-7650 University of Isfahan 2 10.22108/ijgt.2016.21234 20F19 Generalizations of solvable and nilpotent groups Locally graded groups with a condition on infinite subsets Locally graded groups with a condition on infinite subsets Faramarzi Salles Asadollah Damghan University Pazandeh Shanbehbazari Fatemeh Damghan University 01 12 2018 7 4 1 7 17 02 2016 19 06 2016 Copyright © 2018, University of Isfahan. 2018 http://ijgt.ui.ac.ir/article_21234.html

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
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Int. J. Group Theory University of Isfahan International Journal of Group Theory 2251-7650 University of Isfahan 2 10.22108/ijgt.2017.21219 20D15 Nilpotent groups, p-groups Automorphisms of a finite \$p\$-group with cyclic Frattini subgroup Automorphisms of a finite \$p\$-group with cyclic Frattini subgroup Soleimani Rasoul Payame Noor University 01 12 2018 7 4 9 16 08 08 2016 07 01 2017 Copyright © 2018, University of Isfahan. 2018 http://ijgt.ui.ac.ir/article_21219.html

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
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Int. J. Group Theory University of Isfahan International Journal of Group Theory 2251-7650 University of Isfahan 2 10.22108/ijgt.2017.21478 20F05 Generators, relations, and presentations On embedding of partially commutative metabelian groups to matrix groups On embedding of partially commutative metabelian groups to matrix groups Timoshenko E. I. Novosibirsk State Technical University 01 12 2018 7 4 17 26 06 03 2016 29 04 2017 Copyright © 2018, University of Isfahan. 2018 http://ijgt.ui.ac.ir/article_21478.html

‎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
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Int. J. Group Theory University of Isfahan International Journal of Group Theory 2251-7650 University of Isfahan 2 10.22108/ijgt.2017.21479 20F69 Asymptotic properties of groups Measuring cones and other thick subsets in free groups Measuring cones and other thick subsets in free groups Frenkel Elizaveta Moscow State University Remeslennikov Vladimir Mathematical Institute SB RAS 01 12 2018 7 4 27 40 30 03 2016 06 05 2017 Copyright © 2018, University of Isfahan. 2018 http://ijgt.ui.ac.ir/article_21479.html

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
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Int. J. Group Theory University of Isfahan International Journal of Group Theory 2251-7650 University of Isfahan 2 10.22108/ijgt.2017.21610 20D45 Automorphisms The Maschke property for the Sylow \$p\$-sub-groups of the symmetric group \$S_{p^n}\$ The Maschke property for the Sylow \$p\$-sub-groups of the symmetric group \$S_{p^n}\$ Green David J. Institut f&uuml;r Mathematik Friedrich-Schiller&uuml;Universit&auml;t 07737 Jena H&#039;ethelyi ‎L. Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9. Horv&#039;ath E. Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., Department of Algebra, H-1111 Budapest, Műegyetem rkp. 3-9. 01 12 2018 7 4 41 64 23 10 2016 14 08 2017 Copyright © 2018, University of Isfahan. 2018 http://ijgt.ui.ac.ir/article_21610.html

‎‎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
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