University of IsfahanInternational Journal of Group Theory2251-76507220180601Proceedings of Ischia Group Theory 20162452410.22108/ijgt.2018.24524ENJournal Article20200311https://ijgt.ui.ac.ir/article_24524_a37111ac17a4ddc83791c9fb55b3f2f0.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507220180601Sylow multiplicities in finite groups182148210.22108/ijgt.2017.21482ENDanLevyItalyJournal Article20161123Let $G$ be a finite group and let $mathcal{P}=P_{1},ldots,P_{m}$ be a sequence of Sylow $p_{i}$-subgroups of $G$, where $p_{1},ldots,p_{m}$ are the distinct prime divisors of $leftvert Grightvert $. The Sylow multiplicity of $gin G$ in $mathcal{P}$ is the number of distinct factorizations $g=g_{1}cdots g_{m}$ such that $g_{i}in P_{i}$. We review properties of the solvable radical and the solvable residual of $G$ which are formulated in terms of Sylow multiplicities, and discuss some related open questions.https://ijgt.ui.ac.ir/article_21482_4d16a7d4c6f2488422da19da3ac6bcf6.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507220180601Some characterisations of groups in which normality is a transitive relation by means of subgroup embedding properties9162121410.22108/ijgt.2017.21214ENRamonEsteban-RomeroInstitut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Camí de Vera, s/n, 46022 València, Spain0000-0002-2321-8139GiovanniVincenziUniversity of salernoJournal Article20161127In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation in certain universes of groups with some finiteness properties.https://ijgt.ui.ac.ir/article_21214_cc340e8f12a5cd3c8681717909e787b8.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507220180601On finite groups with square-free conjugacy class sizes17242147510.22108/ijgt.2017.21475ENMaria-JoseFelipeUniversidad Politecnica de ValenciaAnaMartinez-PastorUniversidad Politecnica de ValenciaVictor-ManuelOrtiz-SotomayorUniversidad Politecnica de ValenciaJournal Article20161116We report on finite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.https://ijgt.ui.ac.ir/article_21475_3c37cb4c86113971ca7d07fb160d560a.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507220180601On metacyclic subgroups of finite groups25292148010.22108/ijgt.2017.21480ENAdolfoBallester-BolinchesDepartament de Matematiques, Universitat de Valencia, Burjassot, Valencia, Spain0000-0002-2051-9075Journal Article20161027The aim of this survey article is to present some structural results about of groups whose Sylow <em>p</em>-subgroups are metacylic (<em>p</em> a prime). A complete characterisation of non-nilpotent groups whose <em>2</em>-generator subgroups are metacyclic is also presented.https://ijgt.ui.ac.ir/article_21480_b9d4162cb9b5fc9711b0f39d833cf4e0.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507220180601Representations of group rings and groups31442148410.22108/ijgt.2017.21484ENTedHurleyNational University of Ireland GalwayJournal Article20161207An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. It is shown that for any group ring matrix $A$ of $mathbb{C} G$ there exists a matrix $U$ (independent of $A$) such that $U^{-1}AU= diag(T_1,T_2,ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i × s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ are the ranks of the group ring matrices of the primitive idempotents. <br /> <br /> Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping $Amapsto P^{-1}AP$ (fixed $P$) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of $G$ may be derived using the blocks in the images of the group elements. <br /> <br /> For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $mathbb{C}Q$. The characters of $Q$ and the character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This is a special case of the general block diagonalisation process but is arrived at independently. The case for cyclic groups is well-known: Circulant matrices are the group ring matrices of the cyclic group and the Fourier matrix diagonalises any circulant matrix. This has applications to signal processing.https://ijgt.ui.ac.ir/article_21484_2d64c759c091beeacf98923cde8ed7f6.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507220180601On the dimension of the product $[L_2,L_2,L_1]$ in free Lie algebras45502148110.22108/ijgt.2017.21481ENNilMansuroğluAhi Evran UniversityJournal Article20161121Let $L$ be a free Lie algebra of rank $rgeq2$ over a field $F$ and let $L_n$ denote the degree $n$ homogeneous component of $L$. By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field $F$, we determine the dimension of $[L_2,L_2,L_1]$. Moreover, by this method, we show that the dimension of $[L_2,L_2,L_1]$ over a field of characteristic $2$ is different from the dimension over a field of characteristic other than $2$.<br /><br />https://ijgt.ui.ac.ir/article_21481_0b392ad1ffab7cd79272442ecc91712c.pdf