Finite groups with few conjugacy classes of $p$-subgroups

In noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset$\mathcal{C}$stable under inversion. We study the associated Killing formK. For the universal calculus associated to$\mathcal{C}$=G\ {e} we show that the magnitude$\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$of the Killing form is defined for all finite groups (even whenKis not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible,iffμ ≠ 1/(N− 1) whereNis the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N− 1)-dimensional subspace of invariant vectors is invertibleiffthe finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in anySnhas invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groupsD2nwithnodd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

The classification of the finite simple groups was completed sometime during the summer of 1980. To the extent that I can reconstruct things, the last piece in the puzzle was filled in by Ronald Solomon of Ohio State University. At the other chronological extreme, the theory of finite groups can be traced back to its beginnings in the early nineteenth century in the work of Abel, Cauchy, and Galois. Hence the problem of classifying the finite simple groups has a history of over a century and a half. The proof of the Classification Theorem is made up of thousands of pages in various mathematical journals with at least another thousand pages still left to appear in print. Many mathematicians have contributed to the proof; some have spent their entire mathematical lives working on the problem. The problem itself is one of the most natural in mathematics: the group is one of the fundamental structures of modern mathematics; the finite groups are a natural subclass of the class of all groups. Moreover, the finite group theorist is quickly led to consider simple groups via the composition series of a group, and if he is optimistic, to the hope that the finite simple groups might be determined explicitly and much of the structure of the arbitrary finite group retrieved from that of its composition factors. Despite all of this, and despite the fact that most mathematicians learn this much group theory before receiving their Ph.D., the average mathematician does not seem to known much about the classification problem or the mathematics developed to solve it. Within the obvious space limitations of this article, I hope to convey some idea of how the finite simple groups are classified and to relate some of the history of the effort. A more complete description appears in [6], while a very detailed two volume account (by Daniel Gorenstein) is in preparation. A preliminary version of the first quarter of Gorenstein's work appears in [19]. The proceedings of two recent conferences on simple groups containing expository articles on the classification will soon appear in [12] and [13]. Finally an article by Walter Feit on the history of finite group theory through 1961 will appear in [14]. I have included a reasonably lengthy bibliography. Still, many important papers are omitted as they are not directly encountered in the brief outline provided. Other fundamental papers have yet to appear. More complete bibliographies are contained in some of the books mentioned above. Section 1. The Finite Simple Groups

Variants of some of the Brauer-Fowler theorems

A finite group is said to be complete if it has trivial center and if every automorphism is an inner automorphism.A finite group with nonabelian composition factors has a unique completely reducible radical (CR radical).We consider finite groups with nonabelian composition factors whose CR radical consists of complete simple groups and we give necessary and sufficient conditions for such a group to be complete.This involves finding group theoretic conditions which are necessary and sufficient for a finite centerless group to occur as a self-normalizing subgroup of a direct product of symmetric groups. Introduction.A finite group is called semisimple if it contains no nontrivial abelian normal subgroups.A completely reducible group is a group which is isomorphic to a direct product of simple groups.It is well known that a finite semisimple group G contains a unique maximal normal completely reducible group R, called its completely reducible radical.It is easily seen that this completely reducible radical is characteristic in the whole group (Robinson [1, p. 85]).Hence, we have that R < G < Aut R under the obvious embeddings; and this characterizes all semisimple groups with completely reducible radical R. It is an elementary exercise to show that in this situation Aut G = N&ut r(G) where we are identifying G with a subgroup of Aut R in a natural way.In particular, if G is complete, then G = A^Aut r (G).Conversely, a self-normalizing subgroup of Aut R containing R is complete.The above discussion leads to the following proposition (see Robinson [1, P-86]).PROPOSITION 1.There is a bijection between isomorphism classes of finite complete semisimple groups with radical R and conjugacy classes of self-normalizing subgroups of Out R.This follows easily from the above comments and the canonical homomorphism Aut R - Out R.We will be primarily concerned with those finite groups whose composition factors are all nonabelian.We will call these groups purely nonabelian or p.n.a.groups.If R = Dr i (Dr"ij Si) where the Si are nonisomorphic nonabelian finite simple groups, then m OutR Dr((OutSi)wr (Sym(n;))).t=i

According to P. Hall, a subgroup \(H\) of a finite group \(G\) is called pronormal in \(G\) if, for any element \(g\) of \(G\), the subgroups \(H\) and \(H^{g}\) are conjugate in \(\langle H,H^{g}\rangle\). The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group \(G\) in which, for a second maximal subgroup \(H\), its index in \(\langle H,H^{g}\rangle\) does not contain squares for any \(g\) from \(G\). A number of papers by Kondrat’ev, Maslova, Revin, and Vdovin (2012–2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In The Kourovka Notebook, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample \(L_{2}(2^{11})\) to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of second maximal subgroups in the group \(L_{2}(q)\). In addition, for \(q\leq 11\), we find the finite almost simple groups with socle \(L_{2}(q)\) in which all second maximal subgroups are pronormal.

Suppose that G is a finite group, π(G) is the set of prime divisors of its order, and ω(G) is the set of orders of its elements. We define a graph on π(G) with the following adjacency relation: different vertices r and s from π(G) are adjacent if and only if rs ∈ ω(G). This graph is called the Gruenberg-Kegel graph or the prime graph of G and is denoted by GK(G). In a series of papers, we describe the coincidence conditions for the prime graphs of nonisomorphic simple groups. This issue is connected with Vasil’ev’s Question 16.26 in the Kourovka Notebook about the number of nonisomorphic simple groups with the same prime graph. Earlier the author derived necessary and sufficient conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type over fields of orders q and q1, respectively, with the same characteristic. Let G and G1 be two nonisomorphic finite simple groups of Lie type over fields of orders q and q1, respectively, with different characteristics. The author also obtained necessary conditions for the coincidence of the prime graphs of two nonisomorphic finite simple groups of Lie type. In the present paper the latter result is refined in the case where G is a simple linear group of sufficiently high Lie rank over a field of order q. If G is a simple linear group of sufficiently high Lie rank, then we prove that the prime graphs of G and G1 may coincide only in one of the nineteen cases. As corollaries of the main result, we obtain constraints (under some additional conditions) on the possible number of simple groups whose prime graph is the same as the prime graph of a simple linear group.

Let $G$ be a finite group‎. ‎We say that $G$ has emph{spread} r if for any set of distinct non-trivial elements of $G$ $X:={x_1,ldots‎, ‎x_r}subset G^{#}$ there exists an element $yin G$ with the property that $langle x_i,yrangle=G$ for every $1leq ileq r$‎. ‎We say $G$ has emph{exact spread} $r$ if $G$ has spread $r$ but not $r+1$‎. ‎The spreads of finite simple groups and their decorations have been much-studied since the concept was first introduced by Brenner and Wiegold in the mid 1970s‎. ‎Despite this‎, ‎the exact spread of very few finite groups‎, ‎and in particular of the finite simple groups and their decorations‎, ‎is known‎. ‎Here we calculate the exact spread of the sporadic simple Mathieu group M$_{23}$‎, ‎proving that it is equal to 8064‎. ‎The precise value of the exact spread of a sporadic simple group is known in only one other case‎ - ‎the Mathieu group M$_{11}$‎.

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.

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).

According to the classification theorem, the Lyons group <i>Ly</i> is one of the 26 sporadic simple groups and has order 51765179004000000 = 2⁸•3⁷•5⁶•7•11•31•37•67. Since the completion of the classification of all finite simple groups, attention has now turned to other aspects e.g. generations of finite groups which entails determining elements which generate that finite group. As a finite nonabelian simple group, <i>Ly</i> can be generated by a minimum of two of its elements. We thus endeavour in the current study to determine some of the pairs of its elements of distinct prime orders from disctinct conjugacy classes with their product in another conjugacy class of elements of prime order which generate <i>Ly</i> and we call such generations triple generations. Triple generations of any finite group are used in the study of its symmetric genus, where the symmetric genus of a Hurwitz group <I>G</I>, of which <i>Ly</i> is known to be a Hurwitz group, is given by <img width="40" height="25" src="http://article.sciencepublishinggroup.com/journal/147/1471669/image001.png" />. If <I>G</I> is a finite group and <i>lX</i>, <i>mY</i>, <i>nZ</i> are conjugacy classes of elements of <I>G</I>, then <I>G</I> is said to be (<i>l</i>,<i>m</i>,<i>n</i>)-generated if <img width="50" height="10" src="http://article.sciencepublishinggroup.com/journal/147/1471669/image002.png" /> with <i>o</i>(<i>x</i>) = <i>l</i>, <i>o</i>(<i>y</i>) = <i>m</i> and <i>o</i>(<i>xy</i>) = <i>n</i>. The number of distinct ordered pairs (<i>x</i>,<i>y</i>) satisfying <img width="80" height="10" src="http://article.sciencepublishinggroup.com/journal/147/1471669/image003.png" /> such that <i>xy</i> = <i>z</i>, where <img width="40" height="10" src="http://article.sciencepublishinggroup.com/journal/147/1471669/image004.png" /> is an arbitrary class representative, is denoted by <i>ζ<SUB>G</SUB></i>(<i>lX</i>,<i>mY</i>,<i>nZ</i>) and is known as the structure constant of the group algebra <img width="20" height="10" src="http://article.sciencepublishinggroup.com/journal/147/1471669/image005.png" />. The structure constants can be computed from the ordinary character table of <I>G</I>. We shall use the method of the structure constants to determine such generation and/or nongeneration. Thus the object in this paper is to study some of the triple generations of <i>Ly</i> which will thus pave the way towards the study of various combinations of three, four, five etc elements from distinct conjugacy classes which can generate <i>Ly</i> and lead to the ultimate determination of the maximum number of elements of <i>Ly</i> from distinct conjugacy classes of its elements which can generate <i>Ly</i>.

For a finite group $H$‎, ‎let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$‎. ‎In this paper‎, ‎we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$‎, ‎then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$‎. ‎In particular‎, ‎we show that for some finite simple group $S$‎, ‎$G cong S times Z(G)$‎.

The spectrum of a finite group is the set of its element orders. A group is said to be recognizable (by spectrum) if it is isomorphic to any finite group that has the same spectrum. A nonabelian simple group is called quasi-recognizable if every finite group with the same spectrum possesses a unique nonabelian composition factor and this factor is isomorphic to the simple group in question. We consider the problem of recognizability and quasi-recognizability for finite simple groups of types Bn, Cn, and 2Dn with n = 2k.

Let be a finite group and let be a complex irreducible character of . The codegree of is defined by , where is the kernel of . In this paper, we show that if is a finite simple exceptional group of Lie type or a finite simple projective special linear group and is any finite group such that the character codegree sets of and coincide, then and are isomorphic.

We previously have developed two methods (Key–Moori Methods 1 and 2) for constructing codes and designs from finite groups (mostly simple finite groups). In this article, we introduce a new method (Method 3) for constructing codes and designs from fixed points of elements of finite transitive groups. We first discuss background material and results required from finite groups, permutation groups and representation theory. The main aim of this article is to discuss this new method and give some examples by applying it to the sporadic simple groups HS and J 2. In subsequent papers, we aim to apply it to several other simple groups.

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.