University of IsfahanInternational Journal of Group Theory2251-76509420201201On finite-by-nilpotent profinite groups2232292408210.22108/ijgt.2019.119581.1577ENEloisaDetomiDipartimento di Ingegneria dell'Informazione - DEI, Università di Padova,MartaMorigiDipartimento di Matematica, Università di Bologna, Italy.Journal Article20191009Let $gamma_n=[x_1,ldots,x_n]$ 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.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76509420201201Algorithmic problems in Engel groups and cryptographic applications2312502445310.22108/ijgt.2020.119123.1574ENDelaramKahrobaeiDepartment of Computer Science
Deramore Lane, University of YorkMarialauraNoceDepartment of Mathematics (University of Salerno), Italy - Department of Mathematics and Statistic (University of the Basque Country), SpainJournal Article20190916The 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.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76509420201201Engel groups in bath - ten years later2512602442010.22108/ijgt.2020.120132.1584ENAntonioTortoraUniversit&agrave; della Campania &quot;Luigi Vanvitelli&quot; - Caserta - Italy0000-0002-4825-1672MariaTotaUniversità di Salerno - Fisciano - ItalyJournal Article20191122The 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.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76509420201201Groups with many roots2612762449910.22108/ijgt.2020.119870.1582ENSarahHartBirbeck, University of London0000-0003-3612-0736DanielMcVeaghDepartment of Economics, Mathematics and Statistics,
Birkbeck, University of LondonJournal Article20191030Given a prime $p$, a finite group $G$ and a non-identity element $g$, what is the largest number of $p^{th}$ roots $g$ can have? We write $ϱ_p(G)$, or just $ϱ_p$, 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 $ϱ_p$ is large. If there is an element $g$ of $G$ with more $p^{th}$ roots than the identity, then we show $ϱ_p(G) leq ϱ_p(P)$, 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 $ϱ_p(G) leq frac{1}{p}$, while if $G$ is a $p$-group of maximal class, then $ϱ_p(G) leq frac{1}{p} + frac{1}{p^2}$ (both these bounds are sharp). We classify the groups with high values of $ϱ_2$, and give partial results on groups with high values of $ϱ_3$.https://ijgt.ui.ac.ir/article_24499_8550c2e0f9c5a0af741e2b46b04e2ceb.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76509420201201Small doubling in $m$-Engel groups2772912453210.22108/ijgt.2020.121125.1595ENPatriziaLongobardiDepartment of Mathematics,
University of SalernoMercedeMajUniversity of SalernoJournal Article20200115We 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.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76509420201201Some remarks on unipotent automorphisms2933002451910.22108/ijgt.2020.119749.1581ENOrazioPuglisiDipartimento di Matematica, viale Morgagni 67AGunnarTraustasonUniversity of BathJournal Article20191021An 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.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76509420201201Open problems from the conference "Engel Conditions in Groups" held in Bath, UK, 20193013032483010.22108/ijgt.2020.122900.1621ENEdited By GunnarTraustasonUniversity of BathJournal Article20200505Here 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