Classical Groups, Derangements and Primes

Abstract

A classical theorem of Jordan states that every finite transitive permutation group contains a derangement. This existence result has interesting and unexpected applications in many areas of mathematics, including graph theory, number theory and topology. Various generalisations have been studied in more recent years, with a particular focus on the existence of derangements with special properties. Written for academic researchers and postgraduate students working in related areas of algebra, this introduction to the finite classical groups features a comprehensive account of the conjugacy and geometry of elements of prime order. The development is tailored towards the study of derangements in finite primitive classical groups; the basic problem is to determine when such a group G contains a derangement of prime order r, for each prime divisor r of the degree of G. This involves a detailed analysis of the conjugacy classes and subgroup structure of the finite classical groups.

On the Saxl graph of a permutation group

“…First we claim that (a, n) = 1. If n divides q − ǫ then the claim follows from [8,Lemma A.4], so we may assume (n, q − ǫ) = 1. The claim is clear if q is an n-power, so let us assume (n, q) = 1, in which case n divides q n−1 − 1 by Fermat's Little Theorem.…”

Bases for quasisimple linear groups

“…All other elements of prime order in G lie inḠ Z, hence have fixed point space of codimension at least M λ . Hence we see that (12) gives |V | 6 = q 20×6 ≤ |G| · q 14×6 + 2q 21 · q 16×6 , in case (4), q 70×6 ≤ |G| · q 50×6 + 2q 36 · q 60×6 , in case (5).…”

“…, v 5 be an orthonormal basis. Any element of G that fixes the three non-degenerate 2-spaces v 1 , v 2 , v 2 , v 3 and v 3 , v 4 also fixes v 1 , v 5 and v 4 , v 5 (as these are v 2 , v 3 , v 4 ⊥ and v 1 , v 2 , v 3 ⊥ ), hence fixes all the 1-spaces v 1 , . .…”

“…It remains to handle the cases (4), (5), where G may contain graph automorphisms of G. For G 0 = L ǫ 6 (q) or L ǫ 8 (q), the conjugacy classes of involutions in the coset of a graph automorphism are given by [1, §19] for q even, and by [12, 4.5.1] for q odd. It follows that the number of such involutions is less than 2q 21 or 2q 36 in case (4) or (5), respectively.…”

Constructing 2-Arc-Transitive Covers of Hypercubes

“…We say that U is totally singular if Q(u) = 0 for all u ∈ U . The next result is a consequence of Witt's Lemma (see [4]) Proposition 2.1 ([5, Page 38]). Let V be a d-dimensional vector space over the field F q equipped with a nondegenerate quadratic form Q, and U be a maximal totally singular subspace of V .…”