Classification Of Finite Simple Groups Research Articles

A generalization of the Brauer–Fowler theorem

Abstract The famous Brauer–Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most 𝑛 involutions, then the group is finite and its order is bounded in terms of 𝑛 only. This answers a question of Strunkov from the Kourovka notebook.

Dixon's asymptotic without the classification of finite simple groups

AbstractWithout using the classification of finite simple groups (CFSG), we show that the probability that two random elements of generate a primitive group smaller than is at most . As a corollary we get Dixon's asymptotic expansion for the probability that two random elements of (or ) generate a subgroup containing .

On Flag-Transitive, Point-Quasiprimitive Symmetric 2-(v,k,λ) Designs with λ Prime

This paper contributes to the classification of flag-transitive symmetric 2-(v,k,λ) designs with λ prime. We investigate the structure of flag-transitive, point-quasiprimitive automorphism groups (G) of such 2-designs by applying the classification of quasiprimitive permutation groups. It is shown that the automorphism groups (G) have either an abelian socle or a non-abelian simple socle. Moreover, according to the classification of finite simple groups, we demonstrate that point-quasiprimitivity implies point-primitivity of G, except when the socle of G is PSLn(q).

Realizability and tameness of fusion systems

AbstractA saturated fusion system over a finite ‐group is a category whose objects are the subgroups of and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over is realized by a finite group if is a Sylow ‐subgroup of and morphisms in the category are those induced by conjugation in . One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the classification of finite simple groups, but we also give more precise formulations whose proof is independent of the classification.

Bounds for the Diameters of Orbital Graphs of Affine Groups

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer H ≤GL(V ), where the vector space V has dimension d over the prime field, can be bounded in terms of d and log |V|/log |H| only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.

The Significance of Cyclic Groups

Investigation in group theory has been a widespread practice within the realm of mathematics. In finite group theory specifically, mathematicians have historically aimed to classify finite simple groups. Following the turning consensus that the finite simple group classification is complete, research within finite group theory has shifted to other areas. However, this paper endeavors to fill the missing gaps in the current classification – it does not specify the importance of finite group theory. By focusing on introducing, explaining, and demonstrating cyclic groups and their applications, the paper will reveal, specifically, the importance of cyclic groups in the realm of mathematics and beyond. In particular, the qualities of finite cyclic groups allow them to apply easily and broadly from as simple as division rules and binary number systems (by use of a set of integers of modulus n), to more advanced theory and implementation with computer science. Cyclic group theory deserves increased recognition. In addition, by providing definitions, evaluations, proofs, and concrete examples, this paper also endeavors to become a point of reference for readers to understand, consolidate, and further grasp cyclic group theory and its uses.

On oriented m-semiregular representations of finite groups about valency two

Given a group G, an m-Cayley digraph Γ over G is a digraph that has a group of automorphisms isomorphic to G acting semiregularly on the vertex set with m orbits. We say that G admits an oriented m-semiregular representation (OmSR for short), if there exists a regular m-Cayley digraph Γ over G such that Γ is oriented and its automorphism group is isomorphic to G. In particular, O1SR is also named as ORR. Verret and Xia gave a classification of finite simple groups admitting an ORR of valency two in Verret and Xia (2022) [19]. Let m≥2 be an integer. In this paper, we show that all finite groups generated by at most two elements admit an OmSR of valency two except four groups of small orders. Consequently, a classification of finite simple groups admitting an OmSR of valency two is obtained.

Groups of prime degree and the Bateman–Horn Conjecture

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree (qn−1)/(q−1) of PSLn(q) is prime. We present heuristic arguments and computational evidence based on the Bateman–Horn Conjecture to support a conjecture that for each prime n≥3 there are infinitely many primes of this form, even if one restricts to prime values of q. Similar arguments and results apply to the parameters of the simple groups PSLn(q), PSUn(q) and PSp2n(q) which arise in the work of Dixon and Zalesskii on linear groups of prime degree.

An Arad and Fisman’s Theorem on Products of Conjugacy Classes Revisited

A theorem of Z. Arad and E. Fisman establishes that if A and B are two non-trivial conjugacy classes of a finite group G such that either AB=Acup B or AB=A^{-1} cup B, then G cannot be a non-abelian simple group. We demonstrate that, in fact, langle Arangle = langle Brangle is solvable, the elements of A and B are p-elements for some prime p, and langle Arangle is p-nilpotent. Moreover, under the second assumption, it turns out that A=B. This research is done by appealing to recently developed techniques and results that are based on the Classification of Finite Simple Groups.

Second Subgroup of General Linear Group in Dimention 2

In many practical situations we know that a primitive group contains a given permutation and we want to know which group it can be; in some other practical situations we know the group and would like to know if it contains a permutation of some given type. For example, a group G ≤ Sn is said to be non-synchronizing if it is contained in the automorphism group of a non-trivial primitive graph with complete core, that is, with clique number equal to chromatic number. When trying to check if some group is synchronizing, typically, we have only partial information about the graph but enough to say that it has an automorphism of some type, and the goal would be to have in hand a classification of the primitive groups containing permutations of that type. As an illustration of this, the key ingredient in some of the results in [2] was the observation that the primitive graph understudy has a 2-cycle automorphism and hence the automorphism group of the graph is the symmetric group. For many more examples of the importance of knowing the groups that contain permutations of a given type. This type of investigation is certainly very natural since it appears on the eve of group theory, with Jordan, Burnside, Marggraff, but the difficulty of the problem is well illustrated by the very slow progress throughout the twentieth century. Given the new tools available (chiefly the classification of finite simple groups), the topic seems to have new momentum. Let Sn denote the symmetric group on n points; a permutation g∈Sn is said to be imprimitive if there exists an imprimitive group contain in gg. An imprimitive group is said to beminimally imprimitiveifit contains no transitive proper subgroup. An imprimitive group G≤Sn is said to be maximally imprimitive if for all g Sn\G, the group〈g, G 〉 is primitive. The next result, whose proof isstraightforward, provides some alternative characterizations of imprimitive permutations. Now in this paper we discuse Presentation for imprimitive second Subgroup of general linear group in dimention 2 over the field of pk-elements .

An asymptotic for the Hall–Paige conjecture

Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group G of order n satisfying the Hall–Paige condition is (e−1/2+o(1))|Gab|n!2/nn.

Groups of p-rank 2 containing an isolated element of order p

Let p be an odd prime and G be a finite group with O_{p'}(G)=1 of p-rank at most 2 that contains an isolated element of order p. If xnotin Z(G), we show that F^*(G) is simple and we describe the structure of a Sylow p-subgroup P of F^*(G) as well as the fusion system mathcal F_P(F^*(G)) without using the classification of finite simple groups.

Fusion systems with Benson–Solomon components

The Benson–Solomon systems comprise an infinite ascending family of simple exotic fusion systems at the prime 2. The results we prove give significant additional evidence that these are the only simple exotic 2-fusion systems, as conjectured by Solomon. We consider a saturated fusion system F having an involution centralizer with a component C isomorphic to a Benson–Solomon fusion system, and we show under rather general hypotheses that F cannot be simple. Furthermore, we prove that if F is almost simple with these properties, then F is isomorphic to the next larger Benson–Solomon system extended by a group of field automorphisms. Our results are situated within Aschbacher’s program to provide a new proof of a major part of the classification of finite simple groups via fusion systems. One of the most important steps in this program is a proof of Walter’s Theorem for fusion systems, and our first result is specifically tailored for use in the proof of that step. We then apply Walter’s Theorem to treat the general Benson–Solomon component problem under the assumption that each component of an involution centralizer in F is on the list of currently known quasisimple 2-fusion systems.

On the maximal number of elements pairwise generating the symmetric group of even degree

Let G be the symmetric group of degree n. Let ω(G) be the maximal size of a subset S of G such that 〈x,y〉=G whenever x,y∈S and x≠y and let σ(G) be the minimal size of a family of proper subgroups of G whose union is G. We prove that both functions σ(G) and ω(G) are asymptotically equal to 12(nn/2) when n is even. This, together with a result of S. Blackburn, implies that σ(G)/ω(G) tends to 1 as n→∞. Moreover, we give a lower bound of n/5 on ω(G) which is independent of the classification of finite simple groups. We also calculate, for large enough n, the clique number of the graph defined as follows: the vertices are the elements of G and two vertices x,y are connected by an edge if 〈x,y〉≥An.

A characterization of the groups and by their -fusion systems, q odd

Abstract Let q be a nontrivial odd prime power, and let $n \ge 2$ be a natural number with $(n,q) \ne (2,3)$ . We characterize the groups $PSL_n(q)$ and $PSU_n(q)$ by their $2$ -fusion systems. This contributes to a programme of Aschbacher aiming at a simplified proof of the classification of finite simple groups.

Variations on results on orders of products in finite groups

In 2014, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if o(xy) = o(x)o(y) for every x, y ∈ G with (o(x), o(y)) = 1. This has led to a number of results that characterize the nilpotence of a group (or the existence of nilpotent Hall subgroups, or the existence of normal Hall subgroups) in terms of prime divisors of element orders. Here, we look at these results with a new twist. The first of our main results asserts that G is nilpotent if and only if o(xy) ⩽ o(x)o(y) for every x, y ∈ G of prime power order with (o(x), o(y)) = 1. As an immediate consequence, we recover the Baumslag–Wiegold theorem. The proof of this result is elementary. We prove some variations of this result that depend on the classification of finite simple groups.

Baer-Suzuki theorem for the π-radical

In the paper we prove (modulo the classification of finite simple groups) an analogue of the famous Baer-Suzuki theorem for the $\pi$-radical of a finite group, where $\pi$ is a set of primes

Finite simple groups acting with fixity 3 and their occurrence as groups of automorphisms of Riemann surfaces

The results in this article are based on the classification of finite simple groups that act with fixity 3. Motivated by the theory of Riemann surfaces, we investigate which ones of these groups act faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case, we give detailed information about the possible branching data of the surface.

Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups

AbstractIn this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects.

Finite groups contain large centralizers

Every finite non-abelian group of order n has a non-central element whose centralizer has order exceeding n1/3. The proof does not rely on the classification of finite simple groups, yet it uses the Feit-Thompson theorem.