Finite simple characteristic quotients of the free group of rank 2

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.

Suppose that L is a finite group, π(L) is the set of prime divisors of the order |L|, and Y is the class of finite groups G such that π(G) �= π(H) for any proper subgroup H of G. Groups from the class Y will be called prime spectrum minimal. Many but not all finite simple groups are prime spectrum minimal. For finite simple groups not from the class Y, the question whether they are isomorphic to nonabelian composition factors of groups from the class Y is interesting. We describe some finite simple groups that are not isomorphic to nonabelian composition factors of groups from the class Y and construct an example of a finite group from Y that has as its composition factor the finite simple sporadic McLaughlin group Mc L not from the class Y.

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

Let 2 \leq a \leq b \leq c \in \mathbb{N} with \mu=1/a+1/b+1/c<1 and let T=T_{a,b,c}=\langle x,y,z: x^a=y^b=z^c=xyz=1\rangle be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of T ? (Classically, for (a,b,c)=(2,3,7) and more recently also for general (a,b,c) .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove a conjecture of Marion [21] showing that various finite simple groups are not quotients of T , as well as positive results showing that many finite simple groups are quotients of T .

We survey recent progress, made using probabilistic methods, on several problems concerning generation of finite simple groups. For example, we outline a proof that all but finitely many classical groups different from PSp 4 { q ) ( q = 2 a or 3 a ) can be generated by an involution and an element of order 3. Results In this survey we present some new methods and results in the study of generating sets for the finite (nonabelian) simple groups. The results are largely taken from the three papers [16], [17], [18]. We shall present the results in this first section, and outline some proofs in sections 2 and 3. We begin by describing some of the basic questions and work in the area. It is a well known consequence of the classification that every finite simple group can be generated by two elements. This result was established early this century for the alternating groups by Miller [23] and for the groups PSL 2 ( q ) by Dickson [8]. Various other simple groups were handled by Brahana [5] and by Albert and Thompson [1], but it was not until 1962 that Steinberg [26] showed that all finite simple groups of Lie type can be generated by two elements. To complete the picture, in 1984 Aschbacher and Guralnick [2] established the same conclusion for sporadic groups. A refinement of the two element generation question asks whether every finite simple group can be generated by an involution and a further element. Partial results on this question were obtained in the above-mentioned papers [23], [5], [1], [2], but only recently has the question been answered completely, in the affirmative, by Malle, Saxl and Weigel [22].

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.

Amalgams, blocks, weights, fusion systems and finite simple groups

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.

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets (“pseudofree action”) is the alternating group \(\mathbb {A}_5\) acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most \(d\)-dimensional fixed points sets, for a fixed \(d\). We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group \(\mathbb {A}_5\) in dimensions 2, 3 and 5, the linear fractional group \(\mathrm{PSL}_2(7)\) in dimension 5, and possibly the unitary group \(\mathrm{PSU}_3(3)\) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.

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.

The classification of the groups of the title is a natural successor to the classifications of finite simple groups and of locally finite linear simple groups. The statement of the classification is given along with an explanation of the examples. The proof is then discussed.

A minimal permutation representation of a group is its faithful permutation representation of least degree. Here the minimal permutation representations of finite simple exceptional twisted groups are studied: their degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, are found. We can thus assert that, modulo the classification of finite simple groups, the aforesaid parameters are known for all finite simple groups.

Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

In the past two decades, there have been far-reaching developments in the problem of determining all finite non-abelian simple groups—so much so, that many people now believe that the solution to the problem is imminent. And now, as I correct these proofs in October 1980, the solution has just been announced. Of course, the solution will have a considerable effect on many related areas, both within group theory and outside. The purpose of this article is to consider the theory of finite permutation groups with the assumption that the finite simple groups are known, and to examine questions such as: which problems are solved or solvable under this assumption, and what important problems remain?