Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201Codes from $m$-ary $n$-cubes $Q^m_n$: a survey115497510.22060/ajmc.2022.21668.1098ENJennifer D.KeyDepartment of Mathematics, Aberystwyth University, Aberystwyth, SY23 3BZ, UK0000-0002-1487-8127Bernardo G.RodriguesDepartment of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0028, South AfricaJournal Article20220807We collect together some known results concerning the codes from adjacency matrices of the graph with vertices the nodes of the $m$-ary $n$-cube $Q^m_n$ and with adjacency defined by the Lee metric, and include some new results. https://ajmc.aut.ac.ir/article_4975_e42ed06b461ada3f70de5754701bd7ef.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201On a maximal subgroup of the Symplectic group Sp(4,4)1726495510.22060/ajmc.2022.21693.1099ENAyoub Basheer MohammedBasheerSchool of Mathematical and Computer Sciences, University of Limpopo (Turfloop), P Bag X1106, Sovenga 0727, South AfricaJournal Article20220819This paper is dealing with a split extension group of the form 26 :(3× A5), which is the largest maximal subgroup of the Symplectic group Sp(4, 4). We refer to this extension by G. We firstly determine the conjugacy classes of G using the coset analysis technique. The structures of inertia factor groups were determined. We then compute the Fischer matrices of G and apply the Clifford-Fischer theory to calculate the ordinary character table of this group. The Fischer matrices of G are all integer valued, with sizes ranging from 1 to 4. The full character table of G is 26×26 complex valued matrix and is given at the end of this paper.https://ajmc.aut.ac.ir/article_4955_f2630b0b4095ccc2ab8d2836668b1eaf.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201On two generation methods for the simple linear group $PSL(3,7)$2737492910.22060/ajmc.2022.21638.1095ENThekiso TrevorSeretloSchool of Mathematical Sciences, North West University, Mafikeng Branch P/B X2046, Mmabatho 2735, South Africa0000-0003-2834-1942Journal Article20220729A finite group $G$ is said to be \textit{$(l,m, n)$-generated}, if it is a quotient group of the triangle group $T(l,m, n) = \left<x, y, z|x^{l} = y^{m} = z^{n} = xyz = 1\right>.$ In [J. Moori, $(p, q, r)$-generations for the Janko groups $J_{1}$ and $J_{2}$, Nova J. Algebra and Geometry, \bf{2} (1993), no. 3, 277--285], Moori posed the question of finding all the $(p,q,r)$ triples, where $p,\ q$ and $r$ are prime numbers, such that a non-abelian finite simple group $G$ is $(p,q,r)$-generated. Also for a finite simple group $G$ and a conjugacy class $X$ of $G,$ the rank of $X$ in $G$ is defined to be the minimal number of elements of $X$ generating $G.$ In this paper we investigate these two generational problems for the group $PSL(3,7),$ where we will determine the $(p,q,r)$-generations and the ranks of the classes of $PSL(3,7).$ We approach these kind of generations using the structure constant method. GAP [The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.9.3; 2018. (http://www.gap-system.org)] is used in our computations.https://ajmc.aut.ac.ir/article_4929_a2b1bd2dc7f610ad4ece8144155d1ced.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201Reduced designs constructed by Key-Moori Method $2$ and their connection with Method $3$3946500410.22060/ajmc.2022.21378.1092ENAminSaeidiSchool of Mathematical and Computer Sciences, University of Limpopo (Turfloop) Sovenga, South AfricaJournal Article20220706For a 1-$(v,k,\lambda)$ design $\mathcal{D}$ containing a point $x$, we study the set $I_x$, the intersection of all blocks of $\mathcal{D}$ containing $x$. We use the set $I_x$ together with the Key-Moori Method 2 to construct reduced designs invariant under some families of finite simple groups. We also show that there is a connection between reduced designs constructed by Method 2 and the new Moori Method 3.https://ajmc.aut.ac.ir/article_5004_325b9b25249b947a3da5bc02cb7700f3.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201Designs from maximal subgroups and conjugacy classes of $\mathrm{PSL}(2,q)$, $q$ odd4755503410.22060/ajmc.2022.21877.1117ENXavierMbaaleDepartment of Mathematics and Statistics, University of Zambia, Lusaka, ZambiaBernardo G.RodriguesDepartment of Mathematics and Applied Mathematics, University of Pretoria, Hatfield 0028, South AfricaSeiranZandiFarzanegan1 High School, Sanandaj, Kurdistan, IranJournal Article20221023In this paper, using a method of construction of $1$-designs which are not necessarily symmetric, introduced by Key and Moori, we determine a number of $1$-designs with interesting parameters from the maximal subgroups and the conjugacy classes of elements of the group $PSL(2,q)$ for $q$ a power of an odd prime.https://ajmc.aut.ac.ir/article_5034_c5372326408c979c353d9c0f64c69683.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201Some properties of the finite Frobenius groups5761479810.22060/ajmc.2022.21224.1080ENMohammadrezaDarafshehCollege of Science, University of Tehran, IranJournal Article20220316The Frobenius group was defined more than 120 years ago and has been the center of interest for researchers in the field of group theory. This group has two parts, complement and kernel. Proving that the kernel is a normal subgroup has been a challenging problem and several attempts have been done to prove it. In this paper we prove some character theory properties of finite Frobenius groups and also give proofs of normality of the kernel in special cases.https://ajmc.aut.ac.ir/article_4798_de6455af6913b8b1548cbe055c20b16f.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201A generalization of Taketa's theorem on $\rm M$-groups II6367501110.22060/ajmc.2022.21781.1108ENZeinabAkhlaghiDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, IranJournal Article20220917In the recent paper [A generalization of Taketa's theorem on $M$-groups, Quaestiones Mathematicae, (2022)], we give an upper bound $5/2$ for the average of non-monomial character degrees of a finite group $G$, denoted by $\mathrm{acd}_{nm}(G)$, which guarantees the solvability of $G$. Although the result is true, the example we gave to show that the bound is sharp turns out to be incorrect. In this paper we find a new bound and we give an example to show that this new bound is sharp. Indeed, we prove the solvability of $G$, by assuming $\mathrm{acd}_{nm}(G)< \mathrm{acd}_{nm}(\mathrm{SL}_2(5))=19/7$.https://ajmc.aut.ac.ir/article_5011_0057069c3de974695209b73b6eb0947d.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201On the CP exterior product of Lie algebras6978501510.22060/ajmc.2022.21843.1116ENZeinabAraghi RostamiDepartment of Pure Mathematics,
Ferdowsi University of Mashhad, Mashhad, Iran0000-0002-4758-0212Journal Article20221017In this paper, under certain conditions, we show that the non-abelian CP exterior product distributes over direct product of Lie algebras. Then we present some properties about CP extension of Lie algebras.https://ajmc.aut.ac.ir/article_5015_68fe1cc441e0ae46495c34e5d3d5d303.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201Normal supercharacter theory of the dihedral groups7985494510.22060/ajmc.2022.21268.1082ENHadisehSaydiDepartment of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, IranJournal Article20220405Diaconis and Isaacs defined the supercharacter theory for finite groups as a natural generalization of the classical ordinary character theory of finite groups. Supercharacter theory of many finite groups such as the cyclic groups, the Frobenius groups, etc. were well studied and well-known. In this paper we find the normal and automorphic supercharacter theories of the dihedral groups in special cases.https://ajmc.aut.ac.ir/article_4945_68f55a161819b4219be9e47c0de947a7.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201Finite non-solvable groups with few 2-parts of co-degrees of irreducible characters8789499310.22060/ajmc.2022.21894.1119ENNedaAhanjidehDepartment of pure Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P. O. Box 115, Shahrekord, Iran0000-0002-0867-5526Journal Article20221030For a character $ \chi $ of a finite group $ G $, the number $ \chi^c(1)=\frac{[G:{\rm ker}\chi]}{\chi(1)} $ is called the co-degree of $ \chi $. Let ${\rm Sol}(G)$ denote the solvable radical of $G$. In this paper, we show that if $G$ is a finite non-solvable group with $\{\chi^c(1)_2:\chi \in {\rm Irr}(G)\}=\{1,2^m\}$ for some positive integer $m$, then $G/{\rm Sol}(G)$ has a normal subgroup $M/{\rm Sol}(G)$ such that $M/{\rm Sol}(G)\cong {\rm PSL}_2(2^n)$ for some integer $n \geq 2$, $[G:M]$ is odd and $ G/{\rm Sol}(G) \lesssim {\rm Aut}({\rm PSL}_2(2^n))$.https://ajmc.aut.ac.ir/article_4993_0a9857600ac28625b7b6368698f45ead.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201A new characterization of some characteristically simple groups9197486410.22060/ajmc.2022.21283.1083ENZohrehSayanjaliDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran0000-0003-1189-951XJournal Article20220413Let $G$ be a finite group and $\mathrm{cd}(G)$ be the set of irreducible complex character degrees of $G$. It was proved that some finite simple groups are uniquely determined by their orders and their degree graphs. Recently, in [Behravesh, et al., Recognition of Janko groups and some simple $K_4$-groups by the order and one irreducible character degree or character degree graph, Int. J. Group Theory, DOI: 10.22108/ijgt.2019.113029.1502.] new characterizations for some finite simple groups are given. Also, in [Qin, et al., Mathieu groups and its degree prime-power graphs, Comm. Algebra, 2019] the degree prime-power graph of a finite group is introduced and it is proved that the Mathieu groups are uniquely determined by order and degree prime-power graph. In this paper we continue this work and we characterize some simple groups and some characteristically simple groups by their orders and some vertices of their degree prime-power graphs.https://ajmc.aut.ac.ir/article_4864_c466f2fa6e86b0481f688edf99b8852a.pdfAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24494120230201A new approach to character-free proof for Frobenius theorem99103486310.22060/ajmc.2022.21305.1085ENSeyedeh FatemehArfaeezarandiDepartment of Mathematics, Stony Brook University, Stony Brook, New York, USAVahidShahverdiDepartment of Mathematics, KTH Royal Institute of Technology, Stockholm, SwedenJournal Article20220416Let G be a Frobenius group. Using character theory, it is proved that the Frobenius kernel of G is a normal subgroup of G, which is well-known as a Frobenius theorem. There is no known character-free proof for Frobenius theorem. In this note, we prove it, by assuming that Frobenius groups are non-simple. Also, we prove that whether K is a subgroup of G or not, Sylow 2-subgroups of G are either cyclic or generalized quaternion group. Also by assuming some additional arithmetical hypothesis on G we prove Frobenius theorem. We should mention that our proof is character-free.https://ajmc.aut.ac.ir/article_4863_36ef014a4f85c1bbee9f6f358b602a34.pdf