Weak Cayley Table Groups II: Alternating Groups and Finite Coxeter Groups

Abstract

A 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 all finite irreducible Coxeter groups (except possibly E 8) have trivial weak Cayley table group, as well as most alternating groups. We also consider some sporadic simple groups.