On the uniform spread of almost simple symplectic and orthogonal groups

A group is $\frac{3}{2}$-generated if every non-identity element is contained in a generating pair. A conjecture of Breuer, Guralnick and Kantor from 2008 asserts that a finite group is $\frac{3}{2}$-generated if and only if every proper quotient of the group is cyclic, and recent work of Guralnick reduces this conjecture to almost simple groups. In this paper, we prove a stronger form of the conjecture for almost simple symplectic and odd-dimensional orthogonal groups. More generally, we study the uniform spread of these groups, obtaining lower bounds and related asymptotics. This builds on earlier work of Burness and Guest, who established the conjecture for almost simple linear groups.Comment: 32 pages; to appear in J. Algebr

On the uniform domination number of a finite simple group

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep result, we introduce a new invariant $\gamma_u(G)$, which we call the uniform domination number of $G$. This is the minimal size of a subset $S$ of conjugate elements such that for each $1 \ne x \in G$, there exists $s \in S$ with $G = \langle x, s \rangle$. (This invariant is closely related to the total domination number of the generating graph of $G$, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have $\gamma_u(G) \leqslant |C|$ for some conjugacy class $C$ of $G$, and the aim of this paper is to determine close to best possible bounds on $\gamma_u(G)$ for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups $G$ with $\gamma_u(G) = 2$. To do this, we develop a probabilistic approach, based on fixed point ratio estimates. We also establish a connection to the theory of bases for permutation groups, which allows us to apply recent results on base sizes for primitive actions of simple groups.Comment: 35 pages; to appear in Trans. Amer. Math. So

The spread of finite and infinite groups

It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is contained in a generating pair. Much more recently, this result has been generalised in three different directions, which form the basis of this survey article. First, we look at some stronger forms of $\frac{3}{2}$-generation that the finite simple groups satisfy, which are described in terms of spread and uniform domination. Next, we discuss the recent classification of the finite $\frac{3}{2}$-generated groups. Finally, we turn our attention to infinite groups, focusing on the recent discovery that the finitely presented simple groups of Thompson are also $\frac{3}{2}$-generated, as are many of their generalisations. Throughout the article we pose open questions in this area, and we highlight connections with other areas of group theory.Comment: 38 pages; survey article based on my lecture at Groups St Andrews 202

Flexibility in generating sets of finite groups

Let G be a finite group. It has recently been proved that every nontrivial element of G is contained in a generating set of minimal size if and only if all proper quotients of G require fewer generators than G. It is natural to ask which finite groups, in addition, have the property that any two elements of G that do not generate a cyclic group can be extended to a generating set of minimal size. This note answers the question. The only such finite groups are very specific affine groups: elementary abelian groups extended by a cyclic group acting as scalars.Publisher PDFPeer reviewe

The maximal size of a minimal generating set

Funding: The author is a Leverhulme Early Career Fellow, and he thanks the Leverhulme Trust for their support.A generating set for a finite group G is minimal if no proper subset generates G, and m(G) denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist a,b>0 such that any finite group G satisfies m(G)⩽a⋅δ(G)b, for δ(G)=∑p primem(Gp), where Gp is a Sylow p-subgroup of G. To do this, we first bound m(G) for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank 1 or 2). In particular, we prove that there exist a,b>0 such that any finite simple group G of Lie type of rank r over the field Fpf satisfies r+ω(f)⩽m(G)⩽a(r+ω(f))b, where ω(f) denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist a,b>0 such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over Fpf has size at most arb+ω(f).Publisher PDFPeer reviewe

Shintani descent, simple groups and spread

The spread of a group $G$, written $s(G)$, is the largest $k$ such that for any nontrivial elements $x_1, \dots, x_k \in G$ there exists $y \in G$ such that $G = \langle x_i, y \rangle$ for all $i$. Burness, Guralnick and Harper recently classified the finite groups $G$ such that $s(G) > 0$, which involved a reduction to almost simple groups. In this paper, we prove an asymptotic result that determines exactly when $s(G_n) \to \infty$ for a sequence of almost simple groups $(G_n)$. We apply probabilistic and geometric ideas, but the key tool is Shintani descent, a technique from the theory of algebraic groups that provides a bijection, the Shintani map, between conjugacy classes of almost simple groups. We provide a self-contained presentation of a general version of Shintani descent, and we prove that the Shintani map preserves information about maximal overgroups. This is suited to further applications. Indeed, we also use it to study $\mu(G)$, the minimal number of maximal overgroups of an element of $G$. We show that if $G$ is almost simple, then $\mu(G) \leqslant 3$ when $G$ has an alternating or sporadic socle, but in general, unlike when $G$ is simple, $\mu(G)$ can be arbitrarily large.Comment: 30 page

PLACE, IDENTITY, AND LANGUAGE LEARNING: THE TRANSFORMATIVE ROLE OF PLACE-BASED ENGLISH LANGUAGE INSTRUCTION

This study examines the intersections of place and second language learning. Learner identity has been found to be an important construct in second language learning. In recent years, place and space have become central topics in the study of sociolinguistics and identity. One area of place and language that has not been studied in depth, however, is whether place plays a role in second language learning. This study begins to fill this gap by examining the second language learning experiences of thirteen Japanese study abroad students who were enrolled in an eight-week, content-based language course. The content of the course focused on the history and culture of the city in which the course was offered, Memphis, Tennessee. This study demonstrates that the students formed place attachments to the city, that these attachments led to identity shifts, and that the students identity shifts affected their language behavior, identities, and future trajectories. Thirteen Japanese university students between the ages of 18 and 19 took part in the study. Data collection included interviews with students taking the class in 2016 and course alumni from 2012 2015, their social media posts, class blog posts, classwork, and their photographs of Memphis served as the sources of data for this multi-modal study. The participant-provided photographs were also used as an interview elicitation tool. Findings from this study contribute to an understanding of the complexities of place, identity, and language learning. Whereas prior work has pointed to the social capital that can be gained through investment in a second language, this study suggests that investment in place can also lead to gains in social capital. The study also shows that when language learners engage with the history and culture of a place such as Memphis, where racial violence has played such a significant role, that place factors into their future trajectories. Specifically, the participants constructed good language learner and global citizen identities. These findings reveal the power of a place-based curriculum that offers language learners the experience of a fuller spectrum of place and thereby facilitates the difficult work involved in constructing and orienting identity