University of Isfahan International Journal of Group Theory 2251-7650 14 4 2024 11 01 Quantitative characterization of finite simple groups: a complement 181 222 28685 10.22108/ijgt.2024.140804.1893 EN Wujie Shi School of Mathematical Sciences, Suzhou University Suzhou, Jiangsu, P. R. China and School of Mathematics and Big Date, Chongqing University of Arts and Sciences, Chongqing, P. R. China Journal Article 2024 01 24 In this paper, we summarize the research on the characterization of finite simple groups and the study of finite groups based on their ``set of element orders" and ``two orders" (the order of the group and the set of element orders). We also discuss some related topics, their applications, and generalizations. The original version of this paper was published in Chinese in [Scientia Sinica Mathematica, 53 (2023), no. 7, 931--952]. This updated version corrects several errors and includes additional content. It particularly emphasizes the applications of this work in mathematics and computational complexity theory.

https://ijgt.ui.ac.ir/article_28685_0ded34e0ab5cc4a7371098f9030eae4f.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 4 2024 11 11 The probability that two elements of a group have the same centralizers 223 233 28770 10.22108/ijgt.2024.142171.1912 EN Saman Rahimirad Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran Mohammad Zarrin Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX, 78666, USA Journal Article 2024 07 27 In this paper we provide some bounds for the probability, denoted by $\mathcal{PC}(G)$, that two randomly chosen elements in a given finite group have the same centralizers. In particular, among other results, we give the following best possible bounds for $\mathcal{PC}(G)$, depending only on $|G:Z(G)|$ and the smallest prime divisor of $|G|$.

https://ijgt.ui.ac.ir/article_28770_919c859c8fe7d80fc09eef9e7eda022a.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 4 2024 11 11 Lifting automorphisms of subgroups of direct products of cyclic $p$-groups 235 252 28782 10.22108/ijgt.2024.142837.1924 EN Jill Dietz Department of Mathematics, Statistics, and Computer Science, St. Olaf College Northfield, MN, USA Journal Article 2024 09 17 Let $\Gamma$ be a finite group. A subgroup $H$ of $\Gamma$ is called ``fully liftable" in $\Gamma$ if every automorphism of $H$ is the restriction of an automorphism of $\Gamma$. Let $G=C_{p^{k_1}}\times C_{p^{k_2}}$, where $1\le k_1\le k_2$ and $p$ is prime. Using information about the subgroup structure of $G$ and knowledge of ${\rm Aut}(G)$, we characterize all fully liftable subgroups of $G$. It turns out that all cyclic subgroups of $G$ are fully liftable, and non-cyclic subgroups are fully liftable if and only if they are automorphic to certain subproducts of $G$, where two subgroups $H$ and $K$ are automorphic in $G$ if there exists $\alpha\in{\rm Aut}(G)$ such that $\alpha(H)=K$. Further, we compare the fully liftable subgroups of $G$ with the characteristic subgroups of $G$, which are similarly characterized by certain subproducts. Finally, we exhibit some interesting lattice features of both fully liftable subgroups of $G$ and characteristic subgroups of $G$.

https://ijgt.ui.ac.ir/article_28782_fbffada99ad522b1e39bc812b8c844d3.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 4 2024 11 12 Notes on influence of certain permutable subgroups on finite smooth groups 253 262 28872 10.22108/ijgt.2024.142307.1915 EN Mohamed Abd-Ellatif Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, 62511, Beni-Suef, Egypt Journal Article 2024 08 01 A maximal chain of a finite group $G$ is called smooth if any two intervals have the same length are isomorphic. A group $G$ is called totally smooth if all maximal chains of $G$ are smooth, and called generalized smooth if all chains from each subgroup of prime order to $G$ are smooth. In the paper entitled ``Influence of certain permutable subgroups on finite smooth groups" (A. M. Elkholy and A. A. Heliel in Acta Math. Sin. (Engl. Ser.), 27 no. 8 (2011) 1547-1556), the authors investigated the structure of finite groups which have a permutable subgroup of prime order and whose maximal subgroups are totally (or generalized) smooth groups. The results obtained by the authors require further precision. In the proof of some theorems, they overlooked some cases which may represent counterexamples to these theorems. Additionally, in certain theorems, we can omit certain hypotheses and get more accurate results. In this paper, we present counterexamples to some of these results and reintroduce these theorems after modification, using simpler and more direct proofs. Furthermore, we generalize these results by replacing certain hypotheses with weaker ones.

https://ijgt.ui.ac.ir/article_28872_fe96d7765f67d472cf8529ea60c4ac3f.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 4 2024 11 13 Exponential and weakly exponential subgroups of finite groups 263 283 28943 10.22108/ijgt.2024.142395.1918 EN Eric A Swartz Department of Mathematics, William & Mary, Williamsburg, VA 23187, USA Nicholas J. Werner Department of Mathematics, Compueter and Information Science, SUNY at Old Westbury, Old Westbury, NY 11568, USA Journal Article 2024 08 07 Sabatini [L. Sabatini, Products of subgroups, subnormality, and relative orders of elements, Ars Math. Contemp., 24 no. 1 (2024) 9 pp.] defined a subgroup $H$ of $G$ to be an exponential subgroup if $x^{|G:H|} \in H$ for all $x \in G$, in which case we write H ≤exp G. Exponential subgroups are a generalization of normal (and subnormal) subgroups: all subnormal subgroups are exponential, but not conversely. Sabatini proved that all subgroups of a finite group $G$ are exponential if and only if $G$ is nilpotent. The purpose of this paper is to explore what the analogues of a simple group and a solvable group should be in relation to exponential subgroups. We say that an exponential subgroup H ≤exp G is exp-trivial if either $H = G$ or the exponent of $G$, $\exp(G)$, divides $|G:H|$, and we say that a group $G$ is exp-trivial if all exponential subgroups of $G$ are exp-trivial. We classify finite exp-simple groups by proving $G$ is exp-simple if and only if $\exp(G) = \exp(G/N)$ for all proper normal subgroups $N$ of $G$, and we illustrate how the class of exp-simple groups differs from the class of simple groups. Furthermore, in an attempt to overcome the obstacle that prevents all subgroups of a generic solvable group from being exponential, we say that a subgroup $H$ of $G$ is weakly exponential if, for all $x \in G$, there exists $g \in G$ such that $x^{|G:H|} \in H^g$. If all subgroups of $G$ are weakly exponential, then $G$ is wexp-solvable. We prove that all solvable groups are wexp-solvable and almost all symmetric and alternating groups are not wexp-solvable. Finally, we completely classify the groups $PSL(2,q)$ that are wexp-solvable. We show that if $\pi(n)$ denotes the number of primes less than $n$ and $w(n)$ denotes the number of primes $p$ less than $n$ such that $PSL(2,p)$ is wexp-solvable, then \[ \lim_{n \to \infty} \frac{w(n)}{\pi(n)} = \frac{1}{4}.\]

https://ijgt.ui.ac.ir/article_28943_c8e52e2f7205e4062d241aec7647b757.pdf

University of Isfahan International Journal of Group Theory 2251-7650 14 4 2024 11 12 Harada's conjecture II for the finite general linear groups and unitary groups 285 296 28966 10.22108/ijgt.2024.140888.1896 EN Masahiro Sugimoto Department of Mathematics, University of Tsukuba 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577 Japan Journal Article 2024 03 17 K. Harada conjectured for any finite group $G$, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when $G$ is the general linear group over a finite field. We show the conjecture holds if the order of the field is sufficiently large.

https://ijgt.ui.ac.ir/article_28966_3fa1bb244b43c2746b9a7b5d61ebecb0.pdf