Preliminary Results for Groups of Lie Type

Abstract

Let C be a conjugacy class of involutions in a group G. We study the graph Γ(C) whose vertices are elements of C with g, h ∈ C connected by an edge if and only if gh ∈ C. For t ∈ C, we define the component group of t to be the subgroup of G generated by all vertices in Γ(C) that lie in the connected component of the graph that contains t.We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic 2. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic 2 for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group.

The maximal size of a minimal generating set

“…Since , by Lemma 3.1, and, by [12, Lemma 2.8], Now, , so . While the action of on need not be primitive, as explained in the final paragraph of the proof of [10, Corollary 3], we still have , so .…”

“…The author thanks Andrea Lucchini for introducing him to this topic, Nick Gill and Martin Liebeck for information in advance of [10], Colva Roney-Dougal for several influential conversations, particularly concerning Section 3.1, and the anonymous referee for useful comments.…”

“…Much is known about the minimum size of a base, for instance the resolutions of Pyber’s conjecture [9] and Cameron’s conjecture [5], but less is known about and . However, Gill and Liebeck [10] recently proved that any almost simple group of Lie type of rank r over (where p is prime) acting faithfully and primitively on X satisfies the bound . As explained in [10, Example 5.1], must depend on , but Gill and Liebeck conjecture that should only depend on (see [10, Conjecture 5.2] for a precise statement).…”

“…However, Gill and Liebeck [10] recently proved that any almost simple group of Lie type of rank r over (where p is prime) acting faithfully and primitively on X satisfies the bound . As explained in [10, Example 5.1], must depend on , but Gill and Liebeck conjecture that should only depend on (see [10, Conjecture 5.2] for a precise statement). Our final theorem proves this conjecture.…”

Irredundant bases for the symmetric group

“…Recently, Gill and Liebeck showed in [7] that if $$G$$ is an almost simple group of Lie type of rank $$r$$ over the field $$\mathbb {F}_{p^f}$$ of characteristic $$p$$ and $$G$$ is acting primitively, then $$$$\begin{equation*} \operatorname{I}(G, H) \leqslant 177 r^8 + \Omega (f), \end{equation*}$$$$where $$\Omega (f)$$ is the number of prime factors of $$f$$, counted with multiplicity.…”