University of Isfahan International Journal of Group Theory 2251-7650 11 1 2022 03 01 A note on groups with a finite number of pairwise permutable seminormal subgroups 1 6 25337 10.22108/ijgt.2021.119299.1575 EN Alexander Trofimuk Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, Belarus 0000-0003-1262-7401 Journal Article 2019 09 22 A 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, ‎jin {1,ldots‎,‎n}\$‎, ‎\$ineq j\$‎, ‎is studied‎. ‎In particular‎, ‎we prove that if \$G_iin frak F\$ for all \$i\$‎, ‎then \$G^frak Fleq (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.pdf
University of Isfahan International Journal of Group Theory 2251-7650 11 1 2022 03 01 Characterization of the Chevalley group \$G_{2}(5)\$ by the set of numbers of the same order elements 7 16 25484 10.22108/ijgt.2021.120906.1594 EN Maryam Jahandideh Department of Mathematics, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran. Mohammad Reza Darafsheh School of Mathematics, Statistics and computer Science, College of Science, University of Tehran, Tehran, Iran. Journal Article 2020 01 04 Let \$G\$ be a group and \$omega(G)={o(g)|gin G}\$ be the set of element orders of \$G\$. Let \$kinomega(G)\$ and \$s_{k}=|{gin G|o(g)=k}|\$. Let \$nse(G)={s_{k}|kinomega(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 \$Gcong G_{2}(5)\$. https://ijgt.ui.ac.ir/article_25484_884d817927f944f400891de68acb7d1f.pdf
University of Isfahan International Journal of Group Theory 2251-7650 11 1 2022 03 01 The recognition of finite simple groups with no elements of order \$10\$ by their element orders 17 22 25569 10.22108/ijgt.2021.124142.1640 EN Huaiyu He Department of Economics and Management, Shanghai University of Political Science and Law, Shanghai 201701, China Wujie Shi Department of Mathematics, Chongqing University of Arts and Sciences, Chongqing 402160, China3School of Mathe- matics, Suzhou University, Suzhou 215006, China 0000-0002-7352-3764 Journal Article 2020 07 25 The 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.pdf
University of Isfahan International Journal of Group Theory 2251-7650 11 1 2022 03 01 On some projective triply-even binary codes invariant under the Conway group \${rm Co}_1\$ 23 35 25586 10.22108/ijgt.2021.123705.1632 EN Bernardo G. Rodrigues Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa 0000-0002-1349-0219 Journal Article 2020 06 27 A 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.pdf
University of Isfahan International Journal of Group Theory 2251-7650 11 1 2022 03 01 On groups in which subnormal subgroups of infinite rank are commensurable‎ ‎with some normal subgroup 37 42 25599 10.22108/ijgt.2021.127143.1671 EN Ulderico Dardano Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia - Monte S. Angelo, I-80126 Napoli, Italy Fausto De Mari Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università di Napoli “Federico II”, Via Cintia - Monte S. Angelo, I-80126 Napoli, Italy Journal Article 2021 01 24 We 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 \$Hcap 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