University of IsfahanInternational Journal of Group Theory2251-76508220190601On algebraic geometry over completely simple semigroups1102197510.22108/ijgt.2017.21975ENArtem N.ShevlyakovSobolev Institute of MathematicsJournal Article20160416We study equations over completely simple semigroups and describe the coordinate semigroups of irreducible algebraic sets for such semigroups.http://ijgt.ui.ac.ir/article_21975_4fdfd51c66f0672fa3924ddfdc43ad84.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508220190601A classification of nilpotent $3$-BCI groups11242220210.22108/ijgt.2017.100795.1404ENHiroki KoikeNational Autonomous University of MexicoIstvan KovacsUniversity of PrimorskaJournal Article20161205Given a finite group $G$ and a subset $Ssubseteq G,$ the bi-Cayley graph $bcay(G,S)$ is the graph whose vertex set is $G times {0,1}$ and edge set is ${ {(x,0),(s x,1)} : x in G, sin S }$. A bi-Cayley graph $bcay(G,S)$ is called a BCI-graph if for any bi-Cayley graph $bcay(G,T),$ $bcay(G,S) cong bcay(G,T)$ implies that $T = g S^alpha$ for some $g in G$ and $alpha in aut(G)$. A group $G$ is called an $m$-BCI-group if all bi-Cayley graphs of $G$ of valency at most $m$ are BCI-graphs. It was proved by Jin and Liu that, if $G$ is a $3$-BCI-group, then its Sylow $2$-subgroup is cyclic, or elementary abelian, or $Q$ [European J. Combin. 31 (2010) 1257--1264], and that a Sylow $p$-subgroup, $p$ is an odd prime, is homocyclic [Util. Math. 86 (2011) 313--320]. In this paper we show that the converse also holds in the case when $G$ is nilpotent, and hence complete the classification of nilpotent $3$-BCI-groups.http://ijgt.ui.ac.ir/article_22202_277a0945cb23c54a90741a9b98909611.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508220190601${rm B}_pi$-characters and quotients25282220310.22108/ijgt.2017.105879.1442ENMark L.LewisDepartment of Mathematical Sciences
Kent State UniversityJournal Article20170810Let $pi$ be a set of primes, and let $G$ be a finite $pi$-separable group. We consider the Isaacs ${rm B}_pi$-characters. We show that if $N$ is a normal subgroup of $G$, then ${rm B}_pi (G/N) = irr {G/N} cap {rm B}_pi (G)$.http://ijgt.ui.ac.ir/article_22203_a7e18f68145d0f142fed9c528fac1170.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508220190601On free subgroups of finite exponent in circle groups of free nilpotent algebras29402220810.22108/ijgt.2017.108014.1455ENJuliane HansmannDepartment of Mathematics
University of Kiel, Germany0000-0003-0847-7631Journal Article20171115Let $K$ be a commutative ring with identity and $N$ the free nilpotent $K$-algebra on a non-empty set $X$. Then $N$ is a group with respect to the circle composition. We prove that the subgroup generated by $X$ is relatively free in a suitable class of groups, depending on the choice of $K$. Moreover, we get unique representations of the elements in terms of basic commutators. In particular, if $K$ is of characteristic $0$ the subgroup generated by $X$ is freely generated by $X$ as a nilpotent group.http://ijgt.ui.ac.ir/article_22208_1d66e7d97c7526a72c80904843cf42a7.pdfUniversity of IsfahanInternational Journal of Group Theory2251-76508220190601Recognition of the simple groups $PSL_2(q)$ by character degree graph and order41462221210.22108/ijgt.2017.103226.1424ENZeinab AkhlaghiFaculty of Mathematics and Computer science, Amirkabir University of Technology (Tehran
Polytechnic), Tehran, IranMaryam Khatamiorcid.org/0000-0002-4495-8507Behrooz KhosraviFaculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran
Polytechnic), 15914 Tehran, IranJournal Article20170330Let $G$ be a finite group, and $Irr(G)$ be the set of complex irreducible characters of $G$. Let $rho(G)$ be the set of prime divisors of character degrees of $G$. The character degree graph of $G$, which is denoted by $Delta(G)$, is a simple graph with vertex set $rho(G)$, and we join two vertices $r$ and $s$ by an edge if there exists a character degree of $G$ divisible by $rs$. In this paper, we prove that if $G$ is a finite group such that $Delta(G)=Delta(PSL_2(q))$ and $|G|=|PSL_2(q)|$, then $GcongPSL_2(q)$.http://ijgt.ui.ac.ir/article_22212_c6dca60b79cceb6b7922abee8ca0c87c.pdf