University of IsfahanInternational Journal of Group Theory2251-76508420191201The one-prime power hypothesis for conjugacy classes restricted to normal subgroups and quotient groups132300110.22108/ijgt.2018.110074.1472ENJulian BroughFachgruppe Mathematik und Informatik, BU Wuppertal, Wuppertal, GermanyJournal Article20180308We say that a group $G$ satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers. Insoluble groups which satisfy the one-prime power hypothesis have been classified. However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups. In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups.http://ijgt.ui.ac.ir/article_23001_d14fcde9b40ecb50266a73a05dc9479a.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508420191201Further rigid triples of classes in $G_{2}$592300210.22108/ijgt.2018.111467.1481ENMatthew ConderUniversity of CambridgeAlastair LitterickBielefeld University, and Ruhr-University Bochum0000-0002-1355-258XJournal Article20180608We establish the existence of two rigid triples of conjugacy classes in the algebraic group G2 in characteristic 5, complementing results of the second author with Liebeck and Marion. As a corollary, the finite groups G2(5^n) are not (2,4,5)-generated, confirming a conjecture of Marion in this case.http://ijgt.ui.ac.ir/article_23002_15de87ecfb8f838128800886380cf45e.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508420191201Graham Higman's PORC theorem11282300310.22108/ijgt.2018.112574.1498ENMichael Vaughan-LeeOxford University
Mathematical InstituteJournal Article20180817Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments.http://ijgt.ui.ac.ir/article_23003_3a502705d12e7bba8986cf354f103b1a.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508420191201Limits of generalized quaternion groups29362300410.22108/ijgt.2018.112591.1499ENReza HobbiUniversity of TabrizMohammad ShahryariUniversity of TabrizJournal Article20180819In the space of marked groups, we determine the structure of groups which are limit points of the set of all generalized quaternion groups.http://ijgt.ui.ac.ir/article_23004_7f2c4793f60fefcb07ef66735691e602.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508420191201Classifying families of character degree graphs of solvable groups37462300810.22108/ijgt.2018.110140.1473ENMark W.BisslerWestern Governors UniversityJacob LaubacherSt. Norbert College0000-0003-0045-7951Journal Article20180314We investigate prime character degree graphs of solvable groups. In particular, we consider a family of graphs $Gamma_{k,t}$ constructed by adjoining edges between two complete graphs in a one-to-one fashion. In this paper we determine completely which graphs $Gamma_{k,t}$ occur as the prime character degree graph of a solvable group.http://ijgt.ui.ac.ir/article_23008_f4d59d005948621cd6da799f771b59e1.pdf