On the orders of automorphism groups of finite groups. II

John N Bray, Robert A Wilson
2006 Journal of group theroy  
In the Kourovka Notebook, Deaconescu asks if |Aut G| φ(|G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |Aut G| = φ(|G|). In an earlier paper we have answered both questions in the negative, and shown that |Aut G|/φ(|G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble.
doi:10.1515/jgt.2006.036 fatcat:5evkwvjoordnvf2d6ueelf5gwu