University of IsfahanInternational Journal of Group Theory2251-76507320180901On nonsolvable groups whose prime degree graphs have four vertices and one triangle162147610.22108/ijgt.2017.21476ENRoghayehHafeziehDepartment of Mathematics, Gebze Technical University, P.O.Box 41400, Gebze, TurkeyJournal Article20161212Let $G$ 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.https://ijgt.ui.ac.ir/article_21476_7aa9bd067cc2235a1faa46dd8f4728af.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507320180901Groups with permutability conditions for subgroups of infinite rank7162148310.22108/ijgt.2017.21483ENAnna ValentinaDe LucaDipartimento di Matematica e Fisica, Università degli Studi della Campania "Luigi Vanvitelli"RobertoIalentiDipartimento di Matematica e Applicazioni Renato Caccioppoli - Università degli Studi di Napoli Federico IIJournal Article20161128In 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.https://ijgt.ui.ac.ir/article_21483_4b600a56b8f0ea252f47e0a58de19bf7.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507320180901Inertial properties in groups17622161110.22108/ijgt.2017.21611ENUldericoDardanoDipartimento Matematica e Appl., v. Cintia, M.S.Angelo 5a,
I-80126 Napoli (Italy)DikranDikranjanDipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy.SilvanaRinauroSilvana Rinauro, Dipartimento di Matematica, Informatica ed Economia, Universit`a della Basilicata, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy.Journal Article20161201Let $G$ be a group and $p$ be an endomorphism of $G$. A subgroup $H$ of $G$ is called $p$-<em>inert</em> if $H^p\cap H$ 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.https://ijgt.ui.ac.ir/article_21611_00d5ab9d6cd65813b0631a40fa7db9fb.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507320180901Finite groups with non-trivial intersections of kernels of all but one irreducible characters63802160910.22108/ijgt.2017.21609ENMariagraziaBianchiDipartimento di Matematica quot;Federigo Enriques quot;, Università di MilanoMarcelHerzogSchoool of Mathematical Sciences,
Tel-Aviv UniversityJournal Article20160715In 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.https://ijgt.ui.ac.ir/article_21609_42a17a94ecfbfa1359519bb03978b0aa.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507320180901On some integral representations of groups and global irreducibility81942228910.22108/ijgt.2017.100688.1402ENDmitryMalininUWI, Mona, KingstonJournal Article20161130Arithmetic 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 $GL(2,O_K)$.https://ijgt.ui.ac.ir/article_22289_b241fb85a1db50082f5c3c1e8b74e634.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76507320180901Fragile words and Cayley type transducers951092197610.22108/ijgt.2017.100358.1398ENDanieleD'AngeliTUGrazEmanueleRodaroDipartimento di Matematica, Politecnico di Milano, Milano, ItaliaJournal Article20161123We 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.https://ijgt.ui.ac.ir/article_21976_d42f2c0b8452fc83cb7f694995548600.pdf