ON THE ORDERS OF ZEROS OF STRONGLY MONOLITHIC CHARACTERS

Let \(\mathbb F\) be a field of characteristic \(\ell > 0\) and let G be a finite group. It is well-known that the dimension of the minimal projective cover \(\Phi_1^G\) (the so-called 1-PIM) of the trivial left \(\mathbb F[G]\) -module is a multiple of the \(\ell\) -part \(|G|_\ell\) of the order of G. In this note we study finite groups G satisfying \(\dim_{\mathbb F}(\Phi_1^G)=|G|_\ell\) . In particular, we classify the non-abelian finite simple groups G and primes \(\ell\) satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers \(\ell\) (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number \(\ell\in\{2,3,5\}\) implies the existence of an \(\ell^\prime\) -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).

The prime graph of a finite group $G$ is denoted by$ga(G)$. A nonabelian simple group $G$ is called quasirecognizable by primegraph, if for every finite group $H$, where $ga(H)=ga(G)$, thereexists a nonabelian composition factor of $H$ which is isomorphic to$G$. Until now, it is proved that some finite linear simple groups arequasirecognizable by prime graph, for instance, the linear groups $L_n(2)$ and $L_n(3)$ are quasirecognizable by prime graph. In this paper, we consider thequasirecognition by prime graph of the simple group $L_n(5)$.

The Hall graph of a finite group is a simple graph whose vertex set is , the set of all prime divisors of its order, and two distinct primes and are joined by an edge if has at least one Hall -subgroup. For all primes of , we call the -tuple , the degree pattern of Hall graph of , where signifies the degree of vertex . This paper provides some properties of Hall graph. It also gives a characterization for some finite simple groups via order and degree pattern of Hall graph.

Let p be a prime number, k an algebraically closed field of characteristic p, P a finite p-group and W an indecomposable kP-module. Given a finite group G and an indecomposable kG-module V , we say that ðP; W Þ is a vertex–source pair for ðG; V Þ if there is an inclusion P ,! G of groups, under which P is a vertex of V and W is a source of V . Endo-permutation modules occur frequently as sources of simple modules of finite groups. For instance, every simple module of a p-soluble group has endopermutation source; if V is a simple kG-module, where G is a finite group, lying in a nilpotent block of kG, then V has endo-permutation source, and any 2-block of a finite group whose defect groups are isomorphic to the Klein 4-group possesses a simple module with endo-permutation source. By results of Berger and Feit, and independently Puig, the proof of which invokes the classification of finite simple groups, if W is an endo-permutation kP-module which occurs as a source of a simple kG-module, for a p-soluble group G, then the class of W is a torsion element in the Dade group of P. Further, Mazza [8] has shown that any endo-permutation kP-module whose isomorphism class is a torsion element of the Dade group of P and which satisfies certain structural constraints identified by Puig does occur as source of some simple module of some p-nilpotent group. By contrast, the question of which endo-permutation modules occur as sources of simple modules in simple, quasi-simple or almost simple groups has not been extensively studied. The smallest interesting case is the case where P is elementary abelian of rank 2, since if P is a finite cyclic group, there are no non-torsion endo-permutation kPmodules. In this paper, we study two situations in which simple modules of groups related to the finite classical groups having vertex–source pairs ðP; W Þ ,w ithP elementary abelian of order p 2 and W endo-permutation, occur and we prove that in both cases, W must be a self-dual module and hence corresponds to an element of order at most 2 in the corresponding Dade group.

Let be a finite group with , where are prime numbers and are natural numbers. The prime graph of is a simple graph whose vertex set is and two distinct primes and are joined by an edge if and only if has an element of order . The degree of a vertex is the number of edges incident on , and the -tuple is called the degree pattern of . We say that the problem of OD-characterization is solved for a finite group if we determine the number of pairwise non-isomorphic finite groups with the same order and degree pattern as . The purpose of this paper is twofold. First, it completely solves the OD-characterization problem for every finite non-Abelian simple groups their orders having prime divisors at most 17. Second, it provides a list of finite (simple) groups for which the problem of OD-characterization have been already solved.

A finite group $G$ is said to be $(l,m, n)$-generated, if it is a quotient group of the triangle group $T(l,m, n) = left .$ In [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 nonabelian 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 $PSL(3,5),$ where we will determine the $(p,q,r)$-generations and the ranks of the classes of $PSL(3,5).$ 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.

Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 22n+1 and either q − 1, \(q - \sqrt {2q} + 1\) or q + \(\sqrt {2q} + 1\) is a prime number, and M = F4(q), where q = 2 n and either q4 + 1 or q4 − q2 + 1 is a prime number.

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-prime power hypothesis, a group of type($A$),($B$) or ($C$) and certain metacyclic $p-$groups and a minimal non-metacyclic $p-$group where $p$ is a prime number. We will show that the divisibility graph $D(G)$ for all of them has no triangles.

إن لنظرية الزمر وتصنيفاتها أهمية كبيرة في العديد من المجالات الهندسية والفيزيائية والكيميائية، وبشكل خاص بما هو مرتبط بمفهوم التناظر. في هذه المقالة ندرس مسألة توليد زمرة منتهية من بعض الزمر الجزئية منها. فمن أجل الزمر البسيطة المنتهية، بيَّنا أن أي زمرة غير آبلية بسيطة منتهية يمكن توليدها من p_1- زمرة جزئية سيلوفية و p_2-زمرة جزئية سيلوفية، حيث p_1 و p_2 عددين أوليين مختلفين. كما بيَّنا أيضاً أنه من أجل أي عددين أوليين مختلفين p و q، كل زمرة منتهية يمكن توليدها من p-زمرة جزئية سيلوفية و q-زمرة جزئية متداخلة.تتألف المقالة من مقدمة وقسمين أساسيين. ففي أحد الأقسام تمت دراسة توليد الزمر البسيطة المنتهية. وفي القسم الآخر تم ذكر النتائج الأساسية التي تم الحصول عليها والمتعلقة بتوليد الزمر المنتهية من بعض الزمر الجزئية.

In this paper, we study the problem of how a finite group can be generated by some subgroups. In order to the finite simple groups, we show that any finite non-abelian simple group can be generated by two Sylow p1 - and p_2 -subgroups, where p_1  and p_2  are two different primes. We also show that for a given different prime numbers p  and q , any finite group can be generated by a Sylow p -subgroup and a q -subgroup.

We extend two theorems due to P. Flavell [6] to arbitrary fusion systems. The fusion system of a finite group G at a prime p encodes the structure of a Sylow-p-subgroup S together with extra information on G-conjugacy within S in category theoretic terms. From work of Alperin and Broue [1] it emerged that fusion systems of finite groups are particular cases of fusion systems of p-blocks of finite groups. In the early 1990’s, Puig introduced the notion of an abstract fusion system on a finite p-group S as a category whose objects are the subgroups of S and whose morphism sets satisfy a list of properties modelled around what one observes in the case of fusion systems of finite groups and blocks. There are ‘exotic’ fusion systems which do not arise as fusion system of a block. Benson [2] suggested that nonetheless any fusion system should give rise to a p-complete topological space which should coincide with the p-completion of the classifying space BG in case the fusion system does arise as fusion system of a finite group G. Broto, Levi and Oliver laid in [4] the homotopy theoretic foundations of such spaces called p-local finite groups and gave in particular a cohomological criterion for the existence and uniqueness of a p-local finite group associated with a given fusion system. While the existence and uniqueness of p-local finite groups for arbitrary fusion systems is still an open problem, there has been in recent years a steadily growing body of work by many authors trying to add to the understanding of fusion systems by extending classical concepts and results on the p-local structure of finite groups (some of which were relevant for the classification of finite simple groups) to all fusion systems. This is also the underlying philosophy of the present note. The following two theorems are Flavell’s Theorem A and Theorem B in [6], extended to arbitrary fusion systems. Our general terminology on fusion systems follows [7]; in particular, by a fusion system we always mean a saturated fusion system. Theorem 1. Let p be an odd prime, S a finite p-group, F a fusion system on S and α an automorphism of S acting freely on S−{1}, which stabilises F and whose order is a prime number r which does not divide the orders of the automorphism groups AutF(R) for all F-centric radical subgroups R of S. Then F = NF(S). The hypothesis that α stablises F means that for any two subgroups Q, R of S and any morphism φ : Q → R in F the morphism α ◦ φ ◦ α|α(Q) : α(Q) → α(R) is again a morphism in F . Moreover, a subgroup Q of S is called F -centric if CS(Q ) = Z(Q) for all subgroups Q of S such that Q ∼= Q in F ; a subgroup Q of S is called F -radical if Op(AutF (Q)) = AutQ(Q), the group of inner automorphisms of Q. By Alperin’s fusion theorem, F is completely determined by the automorphism groups in F of F -centric radical subgroups of S.

In this paper we prove that a finite group G is isomorphic to the finite projective special unitary group U n ( q ) if and only if they have the same order of Sylow r -normalizer for every prime r .

summary:Let $G$ be a finite group, and let $N(G)$ be the set of conjugacy class sizes of $G$. By Thompson's conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N(G)=N(L)$, then $L $ and $G$ are isomorphic. Recently, Chen et al.\ contributed interestingly to Thompson's conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li's PhD dissertation). In this article, we investigate validity of Thompson's conjecture under a weak condition for the alternating groups of degrees $p+1$ and $p+2$, where $p$ is a prime number. This work implies that Thompson's conjecture holds for the alternating groups of degree $p+1$ and $p+2$.

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

In this article, we are concerned with the nonlinear primitive and monolithic characters of a finite real group. We give some criteria for determining whether or not a finite group is real by using its monolithic characters. Also, we classify all finite real groups which have at most two nonlinear monolithic characters and obtain a result for real groups with three nonlinear monolithic characters. Finally, we prove that there are not any nonlinear primitive irreducible characters of a finite solvable real group having at most five nonlinear monolithic characters.