Invariable generation of permutation and linear groups

Abstract

A subset {x1,x2,…,xd} of a group G invariably generates G if {x1g1,x2g2,…,xdgd} generates G for every d-tuple (g1,g2…,gd)∈Gd. We prove that a finite completely reducible linear group of dimension n can be invariably generated by ⌊[Formula presented]⌋ elements. We also prove tighter bounds when the field in question has order 2 or 3. Finally, we prove that a transitive [respectively primitive] permutation group of degree n≥2 [resp. n≥3] can be invariably generated by O([Formula presented]) [resp. O([Formula presented])] elements.