University of IsfahanInternational Journal of Group Theory2251-765011220220601Variations on Glauberman's ZJ theorem43522560210.22108/ijgt.2021.126329.1659ENDanielAllcockDepartment of Mathematics, University of Texas at Austin, RLM 8.100, 2515 Speedway Stop C1200, Austin, Texas,
USA 78712-12020000-0003-0212-638XJournal Article20201204We give a new proof of Glauberman's ZJ Theorem, in a form that clarifies the choices involved and offers more choices than classical treatments. In particular, we introduce two new ZJ-type subgroups of a $p$-group~$S$, that contain $ZJr(S)$ and $ZJo(S)$ respectively and can be strictly larger.https://ijgt.ui.ac.ir/article_25602_661e6a7a39183188d956b4811280f13f.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011220220601Graphs defined on groups531072560810.22108/ijgt.2021.127679.1681ENPeter J.CameronSchool of Mathematics and Statistics, University of St Andrews, U. K.0000-0003-3130-9505Journal Article20210304This paper concerns aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that, in particular, they are invariant under the action of the automorphism group of $G$). The particular graphs I will chiefly discuss are the power graph, enhanced power graph, deep commuting graph, commuting graph, and non-generating graph.<br />My main concern is not with properties of these graphs individually, but rather with comparisons between them. The graphs mentioned, together with the null and complete graphs, form a hierarchy (as long as $G$ is non-abelian), in the sense that the edge set of any one is contained in that of the next; interesting questions involve when two graphs in the hierarchy are equal, or what properties the difference between them has. I also consider various properties such as universality and forbidden subgraphs,<br />comparing how these properties play out in the different graphs.<br />I have also included some results on intersection graphs of subgroups of various types, which are often in a ''dual'' relation to one of the other graphs considered. Another actor is the Gruenberg--Kegel graph, or prime graph, of a group: this very small graph has a surprising influence over various graphs defined on the group.<br />Other graphs which have been proposed, such as the nilpotence, solvability, and Engel graphs, will be touched on rather more briefly. My emphasis is on finite groups but there is a short section on results for infinite groups. There are briefer discussions of general $Aut(G)$-invariant graphs, and structures other than groups (such as semigroups and rings).<br />Proofs, or proof sketches, of known results have been included where possible. Also, many open questions are stated, in the hope of stimulating further investigation.https://ijgt.ui.ac.ir/article_25608_d761d81a4d488326445a1e3731e0dc74.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011220220601New lower bounds for the number of conjugacy classes in finite nilpotent groups1091192581010.22108/ijgt.2021.128396.1687ENEdward A.BertramDepartment of Mathematics, University of Hawaii, Honolulu, HI 96822, USAJournal Article20210429P. Hall's classical equality for the number of conjugacy classes in $p$-groups yields $k(G) ge (3/2) log_2 |G|$ when $G$ is nilpotent. Using only Hall's theorem, this is the best one can do when $|G| = 2^n$. Using a result of G.J. Sherman, we improve the constant $3/2$ to $5/3$, which is best possible across all nilpotent groups and to $15/8$ when $G$ is nilpotent and $|G| ne 8,16$. These results are then used to prove that $k(G) > log_3(|G|)$ when $G/N$ is nilpotent, under natural conditions on $N trianglelefteq G$. Also, when $G'$ is nilpotent of class $c$, we prove that $k(G) ge (log |G|)^t$ when $|G|$ is large enough, depending only on $(c,t)$.https://ijgt.ui.ac.ir/article_25810_6cf96773007f3bab56c3708c2139b4e7.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011220220601A characterization of GVZ groups in terms of fully ramified characters1211242575210.22108/ijgt.2021.127210.1673ENShawn T.BurkettDepartment of Mathematical Sciences, Kent State University Kent, Ohio 44242, U.S.A.Mark L.LewisDepartment of Mathematical Sciences, Kent State University Kent, Ohio 44242, U.S.A.Journal Article20210128In this paper, we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients. This characterization is based on counting formulas due to Gallagher.https://ijgt.ui.ac.ir/article_25752_ebaa4538ee3863bb82760be6d7cc126f.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011220220601Nullstellensatz for relative existentially closed group1251302570210.22108/ijgt.2021.125453.1652ENMohammadShahryariDepartment of Mathematics, College of Science, Sultan Qaboos University, Muscat, OmanJournal Article20201016We prove that in every variety of $G$-groups, every $G$-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize <strong>Theorem G</strong> of [<em>J. Algebra,</em> <strong>219</strong> (1999) 16--79]. As a result we see that every pair of $G$-existentially closed elements in an arbitrary variety of $G$-groups generate the same quasi-variety and if both of them are $q_{omega}$-compact, they are geometrically equivalent.https://ijgt.ui.ac.ir/article_25702_c4dfb33866e9635cd387f013c94619a8.pdf