University of Isfahan International Journal of Group Theory 2251-7650 9 4 2020 12 01 On finite-by-nilpotent profinite groups 223 229 24082 10.22108/ijgt.2019.119581.1577 EN Eloisa Detomi Dipartimento di Ingegneria dell&#039;Informazione - DEI, Universit&agrave; di Padova, Marta Morigi Dipartimento di Matematica, Universit&agrave; di Bologna, Italy. Journal Article 2019 10 09 Let \$gamma_n=[x_1,ldots,x_n]\$ be the \$n\$th lower central word‎. ‎Suppose that \$G\$ is a profinite group‎ ‎where the conjugacy classes \$x^{gamma_n(G)}\$ contains less than \$2^{aleph_0}\$‎ ‎elements‎ ‎for any \$x in G\$‎. ‎We prove that then \$gamma_{n+1}(G)\$ has finite order‎. ‎This generalizes the much celebrated‎ ‎theorem of B‎. ‎H‎. ‎Neumann that says that the commutator subgroup of a BFC-group is finite‎. ‎Moreover‎, ‎it implies that‎ ‎a profinite group \$G\$ is finite-by-nilpotent if and only if there is a positive integer \$n\$ such that‎ ‎\$x^{gamma_n(G)}\$ contains less than \$2^{aleph_0}\$‎ ‎elements‎, ‎for any \$xin G\$‎. https://ijgt.ui.ac.ir/article_24082_41ac893d8fe29fcd71cc231d5864a1ef.pdf
University of Isfahan International Journal of Group Theory 2251-7650 9 4 2020 12 01 Algorithmic problems in Engel groups and cryptographic applications 231 250 24453 10.22108/ijgt.2020.119123.1574 EN Delaram Kahrobaei Department of Computer Science Deramore Lane, University of York Marialaura Noce Department of Mathematics (University of Salerno), Italy - Department of Mathematics and Statistic (University of the Basque Country), Spain Journal Article 2019 09 16 ‎The theory of Engel groups plays an important role in group theory since these groups are closely related to the Burnside problems‎. ‎In this survey we consider several classical and novel algorithmic problems for Engel groups and propose several open problems‎. ‎We study these problems with a view towards applications to cryptography‎. https://ijgt.ui.ac.ir/article_24453_aa8791fc06eb62b0b6953499acd9f232.pdf
University of Isfahan International Journal of Group Theory 2251-7650 9 4 2020 12 01 Engel groups in bath - ten years later 251 260 24420 10.22108/ijgt.2020.120132.1584 EN Antonio Tortora Universit&amp;agrave; della Campania &amp;quot;Luigi Vanvitelli&amp;quot; - Caserta - Italy 0000-0002-4825-1672 Maria Tota Universit&agrave; di Salerno - Fisciano - Italy Journal Article 2019 11 22 The eighth edition of the international series of Groups St Andrews conferences was held at the University of Bath in 2009 and one of the theme days was dedicated to Engel groups. Since then much attention has been devoted to a verbal generalization of Engel groups. In this paper we will survey the development of this investigation during the last decade. https://ijgt.ui.ac.ir/article_24420_1663b79a0ee9cb6de3ac25a96c0f9507.pdf
University of Isfahan International Journal of Group Theory 2251-7650 9 4 2020 12 01 Groups with many roots 261 276 24499 10.22108/ijgt.2020.119870.1582 EN Sarah Hart Birbeck, University of London 0000-0003-3612-0736 Daniel McVeagh Department of Economics, Mathematics and Statistics, Birkbeck, University of London Journal Article 2019 10 30 Given a prime \$p\$‎, ‎a finite group \$G\$ and a non-identity element \$g\$‎, ‎what is the largest number of \$pth\$ roots \$g\$ can have? We write \$myro_p(G)\$‎, ‎or just \$myro_p\$‎, ‎for the maximum value of \$frac{1}{|G|}|{x in G‎: ‎x^p=g}|\$‎, ‎where \$g\$ ranges over the non-identity elements of \$G\$‎. ‎This paper studies groups for which \$myro_p\$ is large‎. ‎If there is an element \$g\$ of \$G\$ with more \$pth\$ roots than the identity‎, ‎then we show \$myro_p(G) leq myro_p(P)\$‎, ‎where \$P\$ is any Sylow \$p\$-subgroup of \$G\$‎, ‎meaning that we can often reduce to the case where \$G\$ is a \$p\$-group‎. ‎We show that if \$G\$ is a regular \$p\$-group‎, ‎then \$myro_p(G) leq frac{1}{p}\$‎, ‎while if \$G\$ is a \$p\$-group of maximal class‎, ‎then \$myro_p(G) leq frac{1}{p}‎ + ‎frac{1}{p^2}\$ (both these bounds are sharp)‎. ‎We classify the groups with high values of \$myro_2\$‎, ‎and give partial results on groups with high values of \$myro_3\$‎. https://ijgt.ui.ac.ir/article_24499_8550c2e0f9c5a0af741e2b46b04e2ceb.pdf
University of Isfahan International Journal of Group Theory 2251-7650 9 4 2020 12 01 Small doubling in \$m\$-Engel groups 277 291 24532 10.22108/ijgt.2020.121125.1595 EN Patrizia Longobardi Department of Mathematics, University of Salerno Mercede Maj University of Salerno Journal Article 2020 01 15 We study some inverse problems of small doubling type in the class of \$m\$-Engel groups‎. ‎In particular we investigate the structure of a finite subset \$S\$ of a torsion-free \$m\$-Engel group if \$|S^2| = 2|S|+b\$‎, ‎where \$0 leq b leq |S|-4\$‎, ‎for some values of \$b\$‎. https://ijgt.ui.ac.ir/article_24532_a12d90ba9649fee918f3eeb64b1762fc.pdf
University of Isfahan International Journal of Group Theory 2251-7650 9 4 2020 12 01 Some remarks on unipotent automorphisms 293 300 24519 10.22108/ijgt.2020.119749.1581 EN Orazio Puglisi Dipartimento di Matematica, viale Morgagni 67A Gunnar Traustason University of Bath Journal Article 2019 10 21 An automorphism \$alpha\$ of the group \$G\$ is said to be \$n\$-unipotent if \$[g,_nalpha]=1\$ for all \$gin G\$‎. ‎In this paper we obtain some results related to nilpotency of groups of \$n\$-unipotent automorphisms of solvable groups‎. ‎We also show that‎, ‎assuming the truth of a conjecture about the representation theory of solvable groups raised by P‎. ‎Neumann‎, ‎it is possible to produce‎, ‎for a suitable prime \$p\$‎, ‎an example of a f.g‎. ‎solvable group possessing a group of \$p\$-unipotent automorphisms which is isomorphic to an infinite Burnside group‎. ‎Conversely we show that‎, ‎if there exists a f.g‎. ‎solvable group \$G\$ with a non nilpotent \$p\$-group \$H\$ of \$n\$-automorphisms‎, ‎then there is such a counterexample where \$n\$ is a prime power and \$H\$ has finite exponent‎. https://ijgt.ui.ac.ir/article_24519_ae1db50aa114db6fb769bf2702ed6e0c.pdf
University of Isfahan International Journal of Group Theory 2251-7650 9 4 2020 12 01 Open problems from the conference ‎"Engel Conditions in Groups‎" ‎held in Bath‎, ‎UK‎, ‎2019 301 303 24830 10.22108/ijgt.2020.122900.1621 EN Edited By Gunnar Traustason University of Bath Journal Article 2020 05 05 Here is list of open problems from the conference Engel Type Conditions in Groups in Bath that was held in April 2019. https://ijgt.ui.ac.ir/article_24830_1b6f50c9ea660dfb8238ba6e6f292281.pdf