جستجوی مقالات مرتبط با کلیدواژه « conjugacy classes » در نشریات گروه « علوم پایه »

The Conway group $Co_{1}$ is one of the $26$ sporadic simple groups. It is the largest of the three Conway groups with order $4157776806543360000=2^{21}.3^9.5^4.7^2.11.13.23$ and has $22$ conjugacy classes of maximal subgroups. In this paper, we discuss a group of the form $\overline{G}=N\colon G$, where $N=2^{11}$ and $G=M_{24}$. This group $\overline{G}=N\colon G=2^{11}\colon M_{24}$ is a split extension of an elementary abelian group $N=2^{11}$ by a Mathieu group $G=M_{24}$. Using the computed Fischer matrices for each class representative $g$ of $G$ and ordinary character tables of the inertia factor groups of $G$, we obtain the full character table of $\overline{G}$. The complete fusion of $\overline{G}$ into its mother group $Co_1$ is also determined using the permutation character of $Co_1$.

The Mathieu group M24 has a maximal subgroup of the form G ̅=N:G, where N=26 and G=3. S6 ≅ 3. PGL2 (9). Using Atlas, we can see that M24 has only one maximal subgroup of type 26:(3. S6). The group is a split extension of an elementary abelian group, N=26 by a non-split extensionmgroup, G=3. S6. The Fischer matrices for each class representative of G are computed which together with character tables of inertia factor groups of G lead to the full character table of G ̅. The complete fusion of G ̅ into the parent group M24 has been determined using the technique of set intersections of characters.

A finite group G is said to be (l,m, n)-generated, if it is a quotient group of the triangle group T(l,m, n) = ⟨x, y, z| x l =y m = z n= xyz = 1⟩. In [J. Moori, (p, q, r)-generations for the Janko groups J1 and J2, Nova J. Algebra and Geometry, 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.

We continue the investigation, that began in [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20--25.] and [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039--1056.], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm cs}(G)$. Finite groups satisfying ${\rm cs}(G) = \{1, 2, 4, 6\}$ and $\{1, 2, 4, 6, 8\}$ are classified in [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039--1056.] and [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20--25.], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha+1}, 2^{\alpha}3 \}$ if and only if $\alpha =1$. Furthermore, there exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha +1}, 2^{\alpha}3, 2^{\alpha +2}\}$ and $\alpha$ is odd if and only if $\alpha=1$.

Let $K$ be a field and $n\geq 3$. Let $E_n(K)\leq H\leq GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,\ldots,i_m\in\{\pm 1\}$ such that the product $C^{i_1}\cdots C^{i_m}$ contains all nontrivial elementary transvections. In this article we obtain a sharp upper bound for $m(C)$. Moreover, we determine $m(C)$ for any noncentral $H$-class $C$ under the assumption that $K$ is algebraically closed or $n=3$ or $n=\infty$.

A finite group G is called \textit{(l,m,n)-generated}, if it is a quotient group of the triangle group T(l,m,n)=⟨x,y,z|xl=ym=zn=xyz=1⟩. In 29, 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 a (p,q,r)-generated. In this paper we establish all the (p,q,r)-generations of the symplectic group Sp(6,2). GAP 20 and the Atlas of finite group representations 33 are used in our computations.

‎We summarize several results about non-simplicity‎, ‎solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes‎. ‎We also collect some problems that have only been partially solved‎.

We say that a group $G$ satisfies the one-prime power hypothesis for conjugacy classes if the greatest common divisor for all pairs of distinct conjugacy class sizes are prime powers‎. ‎Insoluble groups which satisfy the one-prime power hypothesis have been classified‎. ‎However it has remained an open question whether the one-prime power hypothesis is inherited by normal subgroups and quotients groups‎. ‎In this note we construct examples to show the one-prime power hypothesis is not necessarily inherited by normal subgroups or quotient groups‎.

‎Let $G$ be a finite group‎. ‎We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $chi$ of $G$ such that $chi(g)=0$‎. ‎In this paper‎, ‎we classify groups whose set of vanishing elements is exactly a conjugacy class‎.

ýLet G be a finite group and X be a conjugacy class of G. Theý ýrank of X in G, denoted by rank(G:X), is defined toý ýbe the minimal number of elements of X generating G. In thisý ýpaper we establish the ranks of all the conjugacy classes ofý ýelements for simple alternating group A10 using the structureý ýconstants method and other results established iný ý[A.B.Mý. ýBasheer and Jý. ýMooriý, ýOn the ranks of the alternating group Aný, ýBullý. ýMalaysý. ýMathý. ýSciý. ýSoc..

We report on fi nite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.

We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an influence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau's Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions.

In this paper we classify all finite solvable groups satisfying the following property P5: their orders of representatives are set-wise relatively prime for any 5 distinct non-central conjugacy classes.

For q in {7,8,9,11,13,16}, we consider the primitive actions of L2(q) and use Key-Moori Method 1 as described in [5, 6] to construct designs from the orbits of the point stabilisers and from any union of these orbits. We also use Key-Moori Method 2 (see [8]) to determine the designs from the maximal subgroups and the conjugacy classes of elements of these groups. The incidence matrices of these designs are then used to generate associated binary codes. The full automorphisms of these designs and codes are also determined.

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. Suppose that $ {rm{Irr}} (G | N) $ is the set of the irreducible characters of $G$ that contain $N$ in their kernels. In this paper، we classify solvable groups $G$ in which the set $mathcal {C} (G) = {{rm{Irr}} (G | N) | 1 ne N trianglelefteq G}$ has at most three elements. We also compute the set $mathcal {C} (G) $ for such groups.

A classical result of Neumann characterizes the groups in which each subgroup has finitely many conjugates only as central-by-finite groups. If X is a class of groups, a group G is said to have X-conjugate classes of subgroups if G/coreG(NG(H)) 2 X for each subgroup H of G. Here we study groups which have soluble minimax conjugate classes of subgroups,
giving a description in terms of G/Z(G). We also characterize FC-groups which have soluble minimax conjugate classes of subgroups.