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International Journal of Group Theory - Volume:9 Issue: 4, Dec 2020

International Journal of Group Theory
Volume:9 Issue: 4, Dec 2020

  • تاریخ انتشار: 1399/05/14
  • تعداد عناوین: 7
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  • Eloisa Detomi, Marta Morigi * Pages 223-229
    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 $xin G$‎.
    Keywords: ‎Conjucagy classes‎, ‎verbal subgroups‎, ‎profinite groups‎, ‎FC-groups
  • Delaram Kahrobaei, Marialaura Noce * Pages 231-250
    ‎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.
    Keywords: ‎Engel elements‎, ‎algorithmic problems‎, ‎cryptography
  • Antonio Tortora *, Maria Tota Pages 251-260
    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.
    Keywords: Engel group, verbal subgroup, residually finite group
  • Sarah Hart *, Daniel Mcveagh Pages 261-276
    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 $myro_3$‎.
    Keywords: ‎$pth$ roots‎, ‎square roots‎, ‎cube roots
  • Patrizia Longobardi, Mercede Maj * Pages 277-291
    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 $b$‎.
    Keywords: ‎Direct, inverse problems‎. ‎Small doubling
  • Orazio Puglisi *, Gunnar Traustason Pages 293-300
    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.
    Keywords: unipotent automorphism, solvable group, Engel element
  • Edited By Gunnar Traustason * Pages 301-303
    Here is list of open problems from the conference Engel Type Conditions in Groups in Bath that was held in April 2019.
    Keywords: Engel, commutator, p-group