Weak Cayley table groups III: PSL(2,q)

Abstract

ABSTRACTA weak Cayley table isomorphism is a bijection φ:G→H of groups such that φ(xy)∼φ(x)φ(y) for all x,y∈G. Here ∼ denotes conjugacy. When G = H the set of all weak Cayley table isomorphisms φ:G→G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x↦x−1. Let 𝒲0(G) = ⟨Aut(G),I⟩≤𝒲(G) and say that G has trivial weak Cayley table group if 𝒲(G) = 𝒲0(G). We show that PSL(2,pn) has trivial weak Cayley table group, where p≥5 is a prime and n≥1.