Infinite 32‐generated groups

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.

A group is $\frac 32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac 32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac 32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups $G_1,\, G_2,\, \ldots$ of the Houghton group $\operatorname {FSym}(\mathbb {Z})\rtimes \mathbb {Z}$. We then show that $G_1$ and $G_2$ are $\frac 32$-generated and investigate the related notion of spread for these groups. We are able to show that they have finite spread at least 2. These are, therefore, the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread at least 2 (other than $\mathbb {Z}$ and the Tarski monsters, which have infinite spread). As a consequence, for each $k\in \{2,\, 3,\, \ldots \}$, we also have that $G_{2k}$ is index $k$ in $G_2$ but $G_2$ is $\frac 32$-generated whereas $G_{2k}$ is not.

Finite simple nonabelian groups G that are not π-closed for some set of primes π but have π-closed maximal subgroups (property (*) for (G, π)) are studied. We give a list L of finite simple groups that contains any group G with the above property (for some π). It is proved that 2 ∉ π for any pair (G, π) with property (*) (Theorem 1). In addition, we specify for any sporadic simple group G from L all sets of primes π such that the pair (G, π) has property (*) (Theorem 2). The proof uses the author’s results on the control of prime spectra of finite simple groups.

On Thompson's Theorem

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.

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H1, …, Hm of some special form such that π(H1)∪…∪π(Hm) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections Hi so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H1,…,Hm} controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.

Let G be a finite 2-generated non-cyclic group. The spread of G is the largest integer k such that for any nontrivial elements x1,…,xk, there exists y G ∈ G such that G = 〈xi,y〉 for all i. The more restrictive notion of uniform spread, denoted u(G), requires y to be chosen from a fixed conjugacy class of G, and a theorem of Breuer, Guralnick and Kantor states that u(G) ⩾ 2 for every non-abelian finite simple group G. For any group with u(G) ⩾ 1, we define the uniform domination number γu(G) of G to be the minimal size of a subset S of conjugate elements such that for each nontrivial x ∈ G there exists y ∈ S with G = 〈x, y〉 (in this situation, we say that S is a uniform dominating set for G). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on γu(G) for all simple groups G.In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups G with γu (G) = 2, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for G. We also establish new results concerning the 2-generation of soluble and symmetric groups, and we present several open problems.

A finite group G with conjugacy class rX is said to be rX-complementary generated if, given an arbitrary x∈G-1, there is a y ∈ rXsuch that G <ce:glyph name=dbnd6/> (x,y). The rX-complementary generation of the simple groups was first introduced by Woldar in [17] to show that every sporadic simple group can be generated by an arbitrary element and another suitable element. It is conjectured in [5] that every finite simple group can be generated in this way. In this paper we investigate the rX-complementary generation of the first three Janko groups in an attemp to further develop the techniques of finding rX-complementary generation of the finite simple groups. As a consequence, we obtained all the(p,q,r)-generations of the Janko group J 3, where p,q,r are distinct primes.

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

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.

We provide the first examples of words in the free group of rank 2 which are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.

Generation and random generation: From simple groups to maximal subgroups

We construct an explicit infinite family of pairwise non‐isomorphic infinite simple groups of type (in particular, they are finitely presented) that act faithfully on the circle by orientation‐preserving homeomorphisms, but that admit neither non‐trivial piecewise affine nor piecewise projective actions on the projective line. Our examples are certain forest‐skein groups which, informally, are a mixture of Richard Thompson's groups with Vaughan Jones' planar algebras.

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

The spectrum ω(G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if H≃G for every finite group H such that ω(H)=ω(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum.