On the orders of automorphism groups of finite groups. II

Abstract

In the Kourovka Notebook, Deaconescu asks if |AutG| > φ(|G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |AutG| = φ(|G|). In an earlier paper we have answered both questions in the negative, and shown that |AutG|/φ(|G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. 1. The question, and general overview Let φ denote the Euler totient function, so that φ(n) is the number of integers m with 1 6 m 6 n such that m and n are coprime, and φ(n) n = r ∏