University of IsfahanInternational Journal of Group Theory2251-765011120220301A note on groups with a finite number of pairwise permutable seminormal subgroups162533710.22108/ijgt.2021.119299.1575ENAlexanderTrofimukDepartment of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, BelarusJournal Article20190922A subgroup $A$ of a group $G$ is called {\it seminormal} in $G$, if there exists a subgroup $B$ such that $G=AB$ and $AX$~is a subgroup of $G$ for every subgroup $X$ of $B$. The group $G = G_1 G_2 \cdots G_n$ with pairwise permutable subgroups $G_1,\ldots,G_n$ such that $G_i$ and $G_j$ are seminormal in~$G_iG_j$ for any $i, j\in \{1,\ldots,n\}$, $i\neq j$, is studied. In particular, we prove that if $G_i\in \frak F$ for all $i$, then $G^\frak F\leq (G^\prime)^\frak N$, where $\frak F$ is a saturated formation and $\frak U \subseteq \frak F$. Here $\frak N$ and $\frak U$~ are the formations of all nilpotent and supersoluble groups respectively, the $\mathfrak F$-residual $G^\frak F$ of $G$ is the intersection of all those normal subgroups $N$ of $G$ for which $G/N \in \mathfrak F$.https://ijgt.ui.ac.ir/article_25337_399e1e4b9aa792c0df9603121f7d79e5.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011120220301Characterization of the Chevalley group $G_{2}(5)$ by the set of numbers of the same order elements7162548410.22108/ijgt.2021.120906.1594ENMaryamJahandidehDepartment of Mathematics, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran.Mohammad RezaDarafshehSchool of Mathematics, Statistics and computer Science, College of Science, University of Tehran, Tehran, Iran.Journal Article20200104Let $G$ be a group and $\omega(G)=\{o(g)|g\in G\}$ be the set of element orders of $G$. Let $k\in\omega(G)$ and $s_{k}=|\{g\in G|o(g)=k\}|$. Let $nse(G)=\{s_{k}|k\in\omega(G)\}.$ In this paper, we prove that if $G$ is a group and $G_{2}(5)$ is the Chevalley simple group of type $G_{2}$ over $GF(5)$ such that $nse(G)=nse(G_{2}(5))$, then $G\cong G_{2}(5)$.https://ijgt.ui.ac.ir/article_25484_884d817927f944f400891de68acb7d1f.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011120220301The recognition of finite simple groups with no elements of order $10$ by their element orders17222556910.22108/ijgt.2021.124142.1640ENHuaiyuHeDepartment of Economics and Management, Shanghai University of Political Science and Law, Shanghai 201701, ChinaWujieShiDepartment of Mathematics, Chongqing University of Arts and Sciences, Chongqing 402160, China3School of Mathe-
matics, Suzhou University, Suzhou 215006, ChinaJournal Article20200725The spectrum of a finite group is the set of its element orders. $H$ is said to be a finite cover of $G$ if $G$ is a homomorphic image of $H$ and $H$ is finite. The main aim of this article is to characterize the finite simple groups with no elements of order 10 by its spectrum among covers. At the same time, above simple groups are completely classified. At last, some results on the recognition by spectrum of above groups are also achieved.https://ijgt.ui.ac.ir/article_25569_756a080256c4fec90519527cd91aec44.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011120220301On some projective triply-even binary codes invariant under the Conway group ${\rm Co}_1$23352558610.22108/ijgt.2021.123705.1632ENBernardo G.RodriguesDepartment of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028,
South AfricaJournal Article20200627A binary triply-even $[98280, 25, 47104]_2$ code invariant under the sporadic simple group ${\rm Co}_1$ is constructed by adjoining the all-ones vector to the faithful and absolutely irreducible 24-dimensional code of length 98280. Using the action of ${\rm Co}_1$ on the code we give a description of the nature of the codewords of any non-zero weight relating these to vectors of types 2, 3 and 4, respectively of the Leech lattice. We show that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${\rm Co}_1$. Moreover, we give a partial description of the nature of the codewords of minimum weight of the dual code.https://ijgt.ui.ac.ir/article_25586_233dec6d6c4b8c12fde36df763208c3f.pdfUniversity of IsfahanInternational Journal of Group Theory2251-765011120220301On groups in which subnormal subgroups of infinite rank are commensurable with some normal subgroup37422559910.22108/ijgt.2021.127143.1671ENUldericoDardanoDipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia - Monte S.
Angelo, I-80126 Napoli, ItalyFaustoDe MariDipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia - Monte S.
Angelo, I-80126 Napoli, ItalyJournal Article20210124We study soluble groups $G$ in which each subnormal subgroup $H$ with infinite rank is commensurable with a normal subgroup, i.e. there exists a normal subgroup $N$ such that $H\cap N$ has finite index in both $H$ and $N$. We show that if such a $G$ is periodic, then all subnormal subgroups are commensurable with a normal subgroup, provided either the Hirsch-Plotkin radical of $G$ has infinite rank or $G$ is nilpotent-by-abelian (and has infinite rank).https://ijgt.ui.ac.ir/article_25599_f96d108b8c4ff490e64b857b094cb800.pdf