The distinguishing number of quasiprimitive and semiprimitive groups

Abstract

The distinguishing number of G Sym(Ω) is the smallest size of a partition of Ω such that only the identity of G fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for GL(2, 3) acting on the eight non-zero vectors of F 2 3 , which has distinguishing number three.

Distinguishing simple groups

“…Cameron et al [7,20] showed that all but finitely many primitive permutation groups G, not containing A n , have D(G) = 2. Recently, similar results have been obtained for quasiprimitive and semiprimitive permutation groups in [10].…”

“…If G is primitive, then by [20], we get the list L of fourteen exceptional simple permutation groups that have no regular set. The distinguishing numbers for these groups are computed in [10]. If G is transitive imprimitive, then as a simple group is quasiprimitive, and by [10, Theorems 2], D(G) = 2.…”

Bases of twisted wreath products

“…Moreover, these results were used by Duyan, Halasi and Maróti in their proof of Theorem 3.1 to show that if G is quasiprimitive, then d(G) 4; see [22,Lemma 2.7]. In fact, Devillers, Morgan and Harper proved [19] that if G is quasiprimitive but not primitive, then d(G) = 2. They also proved that if G is semiprimitive but not quasiprimitive, then d(G)…”

Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

“…Theorem 1.9 is derived from the more detailed Theorem 3.4; the proof of the latter theorem forms the most substantial part of this work and occupies most of § 3. We expect that both theorems will be of use in many reduction problems, for example, Theorem 1.9 has recently been exploited to determine the distinguishing numbers of semiprimitive groups [9].…”