On the orders of automorphism groups of finite groups. II

John N Bray 1 , 1  and Robert A Wilson 2 , 2
  • 1 School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS.
  • 2 School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS.

Abstract

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. The problem remains open when G is supersoluble, or nilpotent.

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The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.

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