University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 3 2018 09 01 On nonsolvable groups whose prime degree graphs have four vertices and one triangle 1 6 EN Roghayeh Hafezieh Department of‎ ‎Mathematics‎, ‎Gebze Technical University‎, ‎P.O.Box 41400, Gebze‎, ‎Turkey roghayeh@gtu.edu.tr 10.22108/ijgt.2017.21476 ‎Let \$G\$ be a finite group‎. ‎The prime degree graph of \$G\$‎, ‎denoted‎ ‎by \$Delta(G)\$‎, ‎is an undirected graph whose vertex set is \$rho(G)\$ and there is an edge‎ ‎between two distinct primes \$p\$ and \$q\$ if and only if \$pq\$ divides some irreducible‎ ‎character degree of \$G\$‎. ‎In general‎, ‎it seems that the prime graphs‎ ‎contain many edges and thus they should have many triangles‎, ‎so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles‎. ‎In this paper we consider the case where for a nonsolvable group \$G\$‎, ‎\$Delta(G)\$ is a connected graph which has only one triangle and four vertices‎. prime degree graph,irreducible character degree,triangle http://ijgt.ui.ac.ir/article_21476.html http://ijgt.ui.ac.ir/article_21476_7aa9bd067cc2235a1faa46dd8f4728af.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 3 2018 09 01 Groups with permutability conditions for subgroups of infinite rank 7 16 EN Anna Valentina De Luca Dipartimento di Matematica e Fisica, Universit&amp;agrave; degli Studi della Campania &amp;quot;Luigi Vanvitelli&amp;quot; annavalentina.deluca@unina2.it Roberto Ialenti Dipartimento di Matematica e Applicazioni Renato Caccioppoli - Universit&agrave; degli Studi di Napoli Federico II roberto.ialenti@unina.it 10.22108/ijgt.2017.21483 In this paper, the structure of non-periodic generalized radical groups of infinite rank whose subgroups of infinite rank satisfy a suitable permutability condition is investigated. Group of infinite rank,almost permutable subgroup,nearly permutable subgroup http://ijgt.ui.ac.ir/article_21483.html http://ijgt.ui.ac.ir/article_21483_4b600a56b8f0ea252f47e0a58de19bf7.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 3 2018 09 01 Inertial properties in groups 17 62 EN Ulderico Dardano Dipartimento Matematica e Appl., v. Cintia, M.S.Angelo 5a, I-80126 Napoli (Italy) dardano@unina.it Dikran Dikranjan Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy. dikran.dikranjan@uniud.it Silvana Rinauro Silvana Rinauro, Dipartimento di Matematica, Informatica ed Economia, Universit`a della Basilicata, Via dell&rsquo;Ateneo Lucano 10, I-85100 Potenza, Italy. silvana.rinauro@unibas.it 10.22108/ijgt.2017.21611 ‎‎Let \$G\$ be a group and \$p\$ be an endomorphism of \$G\$‎. ‎A subgroup \$H\$ of \$G\$ is called \$p\$-<em>inert</em> if \$H^pcap H\$ has finite index in the image \$H^p\$‎. ‎The subgroups that are \$p\$-<em>inert</em> for all inner automorphisms of \$G\$ are widely known and studied in the literature‎, ‎under the name inert subgroups‎.<br /> ‎The related notion of <em>inertial endomorphism</em>‎, ‎namely an endomorphism \$p\$ such that all subgroups of \$G\$ are \$p\$-<em>inert‎</em>, ‎was introduced in cite{DR1} and thoroughly studied in cite{DR2,DR4}‎. ‎The ``dual‎" ‎notion of <em>fully inert subgroup</em>‎, ‎namely a subgroup that is \$p\$-<em>inert</em> for all endomorphisms of an abelian group \$A\$‎, ‎was introduced in cite{DGSV} and further studied in cite{Ch+‎, ‎DSZ,GSZ}‎. ‎The goal of this paper is to give an overview of up-to-date known results‎, ‎as well as some new ones‎, ‎and show how some applications of the concept of inert subgroup fit in the same picture even if they arise in different areas of algebra‎. ‎We survey on classical and recent results on groups whose inner automorphisms are inertial‎. ‎Moreover‎, ‎we show how‎<br /> ‎inert subgroups naturally appear in the realm of locally compact topological groups or locally linearly compact topological vector spaces‎, ‎and can be helpful for the computation of the algebraic entropy of continuous endomorphisms‎. ‎‎commensurable‎,‎inert‎,‎inertial endomorphism‎,‎entropy‎,‎intrinsic entropy‎,‎scale function‎,‎growth‎,‎locally compact group‎,‎locally linearly compact space‎,‎Mahler measure‎,‎Lehmer problem http://ijgt.ui.ac.ir/article_21611.html http://ijgt.ui.ac.ir/article_21611_00d5ab9d6cd65813b0631a40fa7db9fb.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 3 2018 09 01 Finite groups with non-trivial intersections of kernels of all but one irreducible characters 63 80 EN Mariagrazia Bianchi Dipartimento di Matematica quot;Federigo Enriques quot;, Universit&agrave; di Milano mariagrazia.bianchi@unimi.it Marcel Herzog Schoool of Mathematical Sciences, Tel-Aviv University herzogm@post.tau.ac.il 10.22108/ijgt.2017.21609 In this paper we consider finite groups \$G\$ satisfying the following‎ ‎condition‎: ‎\$G\$ has two columns in its character table which differ by exactly one‎ ‎entry‎. ‎It turns out that such groups exist and they are exactly the finite groups‎ ‎with a non-trivial intersection of the kernels of all but one irreducible‎ ‎characters or‎, ‎equivalently‎, ‎finite groups with an irreducible character‎ ‎vanishing on all but two conjugacy classes‎. ‎We investigate such groups‎ ‎and in particular we characterize their subclass‎, ‎which properly contains‎ ‎all finite groups with non-linear characters of distinct degrees‎, ‎which were characterized by Berkovich‎, ‎Chillag and Herzog in 1992‎. ‎Finite groups,Complex characters http://ijgt.ui.ac.ir/article_21609.html http://ijgt.ui.ac.ir/article_21609_42a17a94ecfbfa1359519bb03978b0aa.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 3 2018 09 01 On some integral representations of groups and global irreducibility 81 94 EN Dmitry Malinin UWI, Mona, Kingston dmalinin@gmail.com 10.22108/ijgt.2017.100688.1402 Arithmetic aspects of integral representations of finite groups and their irreducibility are considered with a focus on globally irreducible representations and their generalizations to arithmetic rings. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. Let \$K\$ be a finite extension of the rational number field and \$O_K\$ the ring of integers of \$K\$. Let \$G\$ be a finite subgroup of \$GL(2,K)\$, the group of \$(2 times 2)\$-matrices over \$K\$. We obtain some conditions on \$K\$ for \$G\$ to be conjugate to a subgroup of \$GL(2,O_K)\$. globally irreducible representations,class numbers,genera,Hilbert symbol,torsion points of elliptic curves http://ijgt.ui.ac.ir/article_22289.html http://ijgt.ui.ac.ir/article_22289_b241fb85a1db50082f5c3c1e8b74e634.pdf
University of Isfahan International Journal of Group Theory 2251-7650 2251-7669 7 3 2018 09 01 Fragile words and Cayley type transducers 95 109 EN Daniele D'Angeli TUGraz dangeli@math.tugraz.at Emanuele Rodaro Dipartimento di Matematica, Politecnico di Milano, Milano, Italia emanuele.rodaro@polimi.it 10.22108/ijgt.2017.100358.1398 We address the problem of finding examples of non-bireversible transducers defining free groups, we show examples of transducers with sink accessible from every state which generate free groups, and, in general, we link this problem to the non-existence of certain words with interesting combinatorial and geometrical properties that we call fragile words. By using this notion, we exhibit a series of transducers constructed from Cayley graphs of finite groups whose defined semigroups are free, and thus having exponential growth. Fragile words,Cayley type transducers,automaton groups http://ijgt.ui.ac.ir/article_21976.html http://ijgt.ui.ac.ir/article_21976_d42f2c0b8452fc83cb7f694995548600.pdf