Fixed point ratios for finite primitive groups and applications

Abstract

Let G be a finite primitive permutation group on a set Ω and recall that the fixed point ratio of an element x ∈ G, denoted fpr(x), is the proportion of points in Ω fixed by x. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing fpr(x) with the order of x. Our main theorem classifies the triples (G, Ω, x) as above with the property that x has prime order r and fpr(x) > 1/(r + 1). There are several applications. Firstly, we extend earlier work of Guralnick and Magaard by determining the primitive permutation groups of degree m with minimal degree at most 2m/3. Secondly, our main result plays a key role in recent work of the authors (together with Moretó and Navarro) on the commuting probability of p-elements in finite groups. Finally, we use our main theorem to investigate the minimal index of a primitive permutation group, which allows us to answer a question of Bhargava.

On the commuting probability of p-elements in a finite group

“…Proof. Let C = C K (P ) and note that K = CL in view of the inequality in (2). For any prime q, let L q be a P -invariant Sylow q-subgroup of L, which is contained in a P -invariant Sylow q-subgroup K q of K (see [11,Corollary 3.25]).…”

“…Arguing by induction on |K : L|, we may assume that L is a maximal P -invariant subgroup of K. Then L is normal in K and K/L does not have any proper nontrivial P -invariant subgroups, whence (2) implies that C = C K (P ) = C L (P ). If |C K (y) : C L (y)| = |K : L| for every y ∈ P , then P acts trivially on K/L and thus K = CL, which is incompatible with (2). Therefore, we may assume that P = y is cyclic.…”

“…In the language of permutation groups, this number coincides with the fixed point ratio of x with respect to this action, which explains why the main theorem of [2] will be an important ingredient in the proof of Theorem C.…”

“…However, it is worth noting that our proof of Theorem C does not require the classification if we assume that x normalizes, but does not centralize, some normal p ′ -subgroup of G. This implies that the classification is not required for Theorem A under the assumption that the generalized Fitting subgroup of G is a p ′ -group (and so in particular, if G is p-solvable). In order to handle the general case, we use a recent result of the first two authors [2] on fixed point ratios of elements of prime order in primitive permutation groups (see Theorem 3.4).…”

On the generation of simple groups by Sylow subgroups

“…There is an extensive literature on fixed point ratios for groups of Lie type. This includes general bounds, such as the main results in [13,24], as well as more specialized results in [7,8,9,10,16] for classical groups and [23] for exceptional groups. It will also be convenient to use Magma [3] to handle certain low rank groups defined over small fields, which allows us to compute Σ(x) precisely for all x ∈ G of prime order.…”