ORIGINAL_ARTICLE Locally graded groups with a condition on infinite subsets 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 $x\neq y‎, ‎z$ 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 ‎$G\in\mathcal{L}\Leftrightarrow G\in\mathcal{R}\Leftrightarrow G\in\mathcal{S}\Leftrightarrow G\in\mathcal{C}\Leftrightarrow G\in\mathcal{N}_2\mathcal{F}.$‎ http://ijgt.ui.ac.ir/article_21234_67c122bc31064ada379ba0fa8178aec3.pdf 2018-12-01T11:23:20 2019-05-24T11:23:20 1 7 10.22108/ijgt.2016.21234 ‎Finitely generated groups‎ ‎Residually finite groups‎ ‎Locally graded groups Asadollah Faramarzi Salles faramarzi@du.ac.ir true 1 Damghan University Damghan University Damghan University LEAD_AUTHOR Fatemeh Pazandeh Shanbehbazari fateme.pazandeh@gmail.com true 2 Damghan University Damghan University Damghan University AUTHOR  A. Abdollahi, Finitely generated soluble groups with an Engel condition on infinite subsets, Rend. Sem. Mat. Univ. Padova, 103 (2000) 47–49. 1  A. Abdollahi and B. Taeri, A condition on finitely generated soluble groups, Comm. Algebra, 27 (1999) 5633–5638. 2  C. Delizia, Finitely generated soluble groups with a condition on infinite subsets, Istit. Lombardo Accad. Sci. Lett. Rend. A, 128 (1994) 201–208. 3  C. Delizia, On certain residually finite groups, Comm. Algebra, 24 (1996) 3531-3535. 4  C. Delizia and C. Nicotera, Groups with conditions on infinite subsets, Ischia Group Theory 2006: Proceedings of a Conference in Honor of Akbar Rhmetulla, World Scientific Publishing, Singapore, 2007 46–55. 5  C. Delizia, A. Rhemtulla and H. Smith, Locally graded groups with a nilpotency condition on infinite subsets, J. Austral. Math. Soc. Ser. A, 69 (2000) 415–420. 6  J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic pro-p-groups, London Math. Soc. Lecture Note Series, 157, Cambridge Univ. Press, Cambridge, 1991. 7  A. Faramarzi Salles, Finitely generated soluble groups with a condition on infinite subsets, Bull. Aust. Math. Soc., 87 (2013) 152–157. 8  Y. K. Kim and A. H. Rhemtulla, Weak maximality condition and polycyclic groups, Proc. Amer. Math. Soc., 123 (1995) 711–714. 9  J. C. Lennox and J. Wiegold, Extensions of a problem of Paul Erdös on groups, J. Austral. Math. Soc. Ser. A, 31 (1981) 459–463. 10  P. Longobardi, On locally graded groups with an Engel condition on infinite subsets, Arch. Math. (Basel), 76 (2001) 88–90. 11  B. H. Neumann, A problem of Paul Erdös on groups, J. Austral. Math. Soc. Ser. A, 21 (1976) 467–472. 12  D. J. Robinson, A course in the theory of groups, Second Edition, Springer-Verlag, Berlin, 1982. 13  J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972) 250–270. 14
ORIGINAL_ARTICLE Automorphisms of a finite $p$-group with cyclic Frattini subgroup 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 2018-12-01T11:23:20 2019-05-24T11:23:20 9 16 10.22108/ijgt.2017.21219 ‎‎Automorphism group‎ ‎Finite $p$-group‎ ‎Frattini subgroup‎ Rasoul Soleimani rsoleimanii@yahoo.com true 1 Payame Noor University Payame Noor University Payame Noor University LEAD_AUTHOR  J. E. Adney and T. Yen, Automorphisms of a p-group, Illinois. J. Math., 9 (1965) 137–143. 1  T. R. Berger, L. G. Kovács and M. F. Newman, Groups of prime power order with cyclic Frattini subgroup, Nederl. Acad. Westensch. Indag. Math., 83 (1980) 13–18. 2  A. Caranti and C.M. Scoppola, Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup, Arch. Math., 56 (1991) 218–227. 3  M. J. Curran and D. J. McCaughan, Central automorphisms that are almost inner, Comm. Algebra, 29 (2001) 2081–2087. 4  S. Fouladi, A. R. Jamali and R. Orfi, On the automorphism group of a finite p-group with cyclic Frattini subgroup, Math. Proc. Royal Irish Academy, 108A (2008) 165–175. 5  The GAP Group, GAP-Groups, Algorithms and Programing, Version 4.4, 2005, (http://www.gap-system.org). 6  D. J. Gorenstein, Finite Groups, Chelsea Publishing Company, New York, 1968. 7  B. Huppert, Endliche Gruppen I, Grundlehren der Mathematischen Wissenschaften Springer-Verlag, 134, Berlin, 1967. 8  M. Morigi, On the minimal number of generators of finite non-abelian p-groups having an abelian automorphism group, Comm. Algebra, 23 (1995) 2045–2065. 9  O. Müller, On p-automorphisms of finite p-groups, Arch. Math., 32 (1979) 533–538. 10
ORIGINAL_ARTICLE On embedding of partially commutative metabelian groups to matrix groups ‎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 2018-12-01T11:23:20 2019-05-24T11:23:20 17 26 10.22108/ijgt.2017.21478 Partially commutative group Metabeliah group universal theory Equations in group E. I. Timoshenko eitim45@gmail.com true 1 Novosibirsk State Technical University Novosibirsk State Technical University Novosibirsk State Technical University LEAD_AUTHOR  Ch. K. Gupta and E. I. Timoshenko, Partially Commutative Metabelian Groups: Centralizers and elementary Equa-tion, Algebra Logic, 48 (2009) 173–192. 1  E. I. Timoshenko, Universal Equivalence of Partially Commutative Metabelian Groups, Algebra Logic, 49 (2010) 177–196. 2  Ch. K. Gupta and E. I. Timoshenko, On Universal Theories of Partially Commutative Metabelian Groups, Algebra Logic, 50 (2011) 1–16. 3  E. I. Timoshenko, A Mal’tsev Basis for a Partially Commutative Nilpotent Metabelian Group, Algebra Logic, 50 (2011) 647–658. 4  Ch. K. Gupta and E. I. Timoshenko, Properties and Universal Theories of Partially Commutative Metabelian Nilpotent Groups, Algebra Logic, 51 (2012) 285–305. 5  V. N. Remeslennikov and V. G. Sokolov, Some Properties of the Magnus Embedding, (Russian), Algebra Logic, 9 (1970) 342–349. 6  E. I. Timoshenko, Endomorphisms and Universal Theories of Solvable Groups, Novosibirsk: NSTU publishers, 2013 pp. 327. 7  O. Chapuis, Universal Theory of Certain Solvable Groups and Bounded Ore Group Rings, J. Algebra, 176 (1995) 368–391. 8  Unsolved Probles in Group Theory, The Kourovka Notebook, issue 17, 2011. 9  E. R. Green, Graph products of groups, PhD Thesis of Newcastleupon-Tyne, 2006. 10  L. J. Corredor and M. A. Gutierrez, A generating set for the automorphism group of a graph product of abelian groups, Internat. J. Algebra Comput., 22 (2012) pp. 21, arXiv:0911.0576v1[math.GR], 2009. 11  R. Charney, K. Ruane, N. Stambaugh and A. Vijayan, The automorphisms group of a graph product with no SIL, Illinois J. Math., 54 (2010) 249–262. 12  E. I. Timoshenko, Metabelian Groups with one Defining Relation, and the Magnus Embedding, Math. Notes, 57 (1995) 414–420. 13  A. Myasnikov and P. Shumyatsky, Discriminating groups and c-dimension, J. Group Theory, 7 (2004) 135–142. 14
ORIGINAL_ARTICLE Measuring cones and other thick subsets in free groups 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 2018-12-01T11:23:20 2019-05-24T11:23:20 27 40 10.22108/ijgt.2017.21479 free group ‎$lambda-$measure regular subset special automaton thick monoid Elizaveta Frenkel lizzy.frenkel@gmail.com true 1 Moscow State University Moscow State University Moscow State University LEAD_AUTHOR Vladimir Remeslennikov remesl@ofim.oscsbras.ru true 2 Mathematical Institute SB RAS Mathematical Institute SB RAS Mathematical Institute SB RAS AUTHOR  Ya. S. Averina and E. V. Frenkel, On strictly sparse subsets of a free group, Sib. lektron. Mat. Izv., 2 (2005) 1–13, http://semr.math.nsc.ru. 1  A. V. Borovik, A. G. Myasnikov and V. N. Remeslennikov, Multiplicative measures on free groups, Internat. J. Algebra Comput., 13 no. 6 (2003) 705--731. 2  E. Yu. Daniyarova, A. G. Myasnikov and V. N. Remeslennikov, Dimension in universal algebraic geometry, (Russian), translated from Dokl. Akad. Nauk, 457 no. 3 (2014) 265–267. 3  D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson and W. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992. 4  P. Flajolet and R. Sedgwick, Analytic Combinatorics: Functional Equations, Rational and Algebraic Functions. In: INRIA, RR 4103 2001. 5  E. Frenkel, A. G. Myasnikov and V. N. Remeslennikov, Regular sets and counting in free groups, Combinatorial and Geometric Group Theory, Trends in Mathematics, Birkhauser Verlag, Basel/Switzerland, 2010 93–118. 6  E. Frenkel, A. G. Myasnikov and V. N. Remeslennikov, Amalgamated products of groups: measures of random normal forms, Fundam. Prikl. Mat., 16 no. 8 (2010) 189-221. 7  E. Frenkel and V. N. Remeslennikov, Double cosets in free groups, Internat. J. Algebra Comput., 23 no. 5 (2013) 1225–1241. 8  E. Frenkel and V. N. Remeslennikov, Cones and thick monoids in free groups, Materials of International Workshop ’Almaz-2’, OmGTU press, Omsk, 2015 64–68. 9  R. Gilman, Formal languages and their application to combinatorial group theory, Groups, Languages Algorithms, 1-36, Contemp. Math., 378, Amer. Math. Soc., Providence, RI, 2005. 10  J. G. Kemeny and J. L. Snell, Finite Markov chains, The University Series in Undergraduate Mathematics D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. 11  M. V. Lawson, Finite automata, Chapman & Hall, CRC Press, 2004. 12  M. O. Rabin and D. Scott, Finite automata and their decision problems, IBM J. Res. Develop., 3 13 no. 2 (1959) 114–125. 14
ORIGINAL_ARTICLE The Maschke property for the Sylow $p$-sub\-groups of the symmetric group $S_{p^n}$ ‎‎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 2018-12-01T11:23:20 2019-05-24T11:23:20 41 64 10.22108/ijgt.2017.21610 ‎Maschke's Theorem‎ ‎coprime action‎ ‎Sylow $p$-subgroup of symmetric group‎ ‎iterated wreath product‎ ‎uniserial action David Green david.green@uni-jena.de true 1 Institut f&uuml;r Mathematik Friedrich-Schiller&uuml;Universit&auml;t 07737 Jena Institut f&uuml;r Mathematik Friedrich-Schiller&uuml;Universit&auml;t 07737 Jena Institut f&uuml;r Mathematik Friedrich-Schiller&uuml;Universit&auml;t 07737 Jena AUTHOR ‎L. H&#039;ethelyi fobaba@t-online.hu true 2 Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9. Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9. Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9. AUTHOR E. Horv&#039;ath he@math.bme.hu true 3 Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., 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. Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., Department of Algebra, H-1111 Budapest, Műegyetem rkp. 3-9. LEAD_AUTHOR  Y. 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