University of IsfahanInternational Journal of Group Theory2251-765010220210601The minimum sum of element orders of finite groups55602385810.22108/ijgt.2019.115910.1538ENMaghsoudJahaniDepartment of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, IranYadollahMarefatDepartment of Mathematics, Shabestar branch, Islamic Azad University, Shabestar, Iran0000-0003-1156-6707HasanRefaghatDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, IranBahramVakili FasaghandisiDepartment of Mathematics, Shabestar branch, Islamic Azad University, Shabestar, IranJournal Article20190307Let $ G $ be a finite group and \( \psi(G)=\sum_{g\in G}o(g) \), where $ o(g) $ denotes the order of $g\in G$. We show that the Conjecture 4.6.5 posed in [Group Theory and Computation, (2018) 59-90], is incorrect. In fact, we find a pair of finite groups $G$ and $S$ of the same order such that $ \psi(G)<\psi(S)$, with $G$ solvable and $S$ simple.https://ijgt.ui.ac.ir/article_23858_0ac95daf1e2e870740851f216e4a425f.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765010220210601Hilbert's theorem 90 for finite nilpotent groups61642385710.22108/ijgt.2019.116275.1545ENWilliamCockeDepartment of Mathematics, University of Wisconsin Madison, WI, USAJournal Article20190404In this note we prove an analog of Hilbert's theorem 90 for finite nilpotent groups. Our version of Hilbert's theorem 90 was inspired by the Boston--Bush--Hajir (BBH) heuristics in number theory and will be useful in extending the BBH heuristics beyond quadratic field extensions.https://ijgt.ui.ac.ir/article_23857_466cd68a8d70a69edef9e5971babc3f4.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765010220210601Influence of complemented subgroups on the structure of finite groups65742426110.22108/ijgt.2019.119105.1570ENIzabela AgataMalinowskaFaculty of Mathematics, University of Bialystok, 15-245 Bialystok, Ciolkowskiego 1M, Poland0000-0002-2772-7379Journal Article20190910P. Hall proved that a finite group $G$ is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of $G$ is complemented in $G$. He called such groups complemented. A. Ballester-Bolinches and X. Guo established the structure of minimal non-complemented groups. We give the classification of finite non-soluble groups all of whose second maximal subgroups are complemented groups. We also prove that every finite group with less than $21$ non-complemented non-minimal $\{2,3,5\}$-subgroups is soluble.https://ijgt.ui.ac.ir/article_24261_26abaf2b4bf9f112a26c308aa371615b.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765010220210601On infinite groups whose finite quotients have restricted prime divisors75882451810.22108/ijgt.2020.119054.1569ENDerek J. S.RobinsonDepartment of Mathematics, University of Illinois at Urbana-Champaign Urbana, IL, 61801, USA.Journal Article20190906The effect of restricting the set of primes dividing the orders of the finite quotients of a group is investigated. Particular attention is paid to abelian, soluble, locally soluble and locally finite groups. The connection with the extraction of roots is explored.https://ijgt.ui.ac.ir/article_24518_142b9836e070db7976aa0b32eb22eae6.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765010220210601On the power graphs of elementary abelian and extra special $p$-groups89952453110.22108/ijgt.2020.120552.1588ENMasoudPourhasanDepartment of Mathematics, Science and Research Branch Islamic Azad University, Tehran, IranHosseinDoostieDepartment of Mathematics, Science and Research Branch Islamic Azad University, Tehran, IranJournal Article20191215For a given odd prime $p$, we investigate the power graphs of three classes of finite groups: the elementary abelian groups of exponent $p$, and the extra special groups of exponents $p$ or $p^2$. We show that these power graphs are Eulerian for every $p$. As a corollary, we describe two classes of non-isomorphic groups with isomorphic power graphs. In addition, we prove that the clique graphs of the power graphs of two considered classes are complete.https://ijgt.ui.ac.ir/article_24531_f2605f3fd1e6973bd072461a34d1e78d.pdf