From: mareg@mimosa.csv.warwick.ac.uk () Subject: Re: Bound for the automorphism group Date: 9 Mar 2001 14:35:15 GMT Newsgroups: sci.math Summary: Computable lower bounds for the order of Aut(G) In article , olavesh@netvis.com (Ola Veshta) writes: >>If G is a finite group with n elements is there a lower bound for >>the >>size of its automorphism group Aut(G) ? > >>Thanks, >>Ola > >Is this question so difficult ? I was sure that some group theory >expert will give some answer. You are really just asking somebody else to do a literature search for you, which you could just as easily do for yourself! W. Ledermann and B.H. Neumann proved in "On the Order of the Automorphism Group of a Finite Group I", Proc. Roy. Soc. London Ser. A 233 (1956), 494-506, that there is a function f(n) such that for any n, and group G with |G| >= f(n) has |Aut(G)| >= n. Thye show f(n) <= (n-1)^x, where x = n + floor ( (n-2) log_2 (n-1) ). This was refined by J.A. Green, "On the number of automorphisms of a finite group", Proc. Roy. Soc. London Ser. A 237 (1956), 574-558, who showed that for anu prime p, |Aut(G)| is divisible by p^h, whenever |G| is divisible by p^g, where g = h(h+3)/2 + 1. This was refined again by J.C. Howarh, "On the power of a prime dividing the order of the automorphism group of a finite group", Proc. Glasgow Math. Assoc. 4, 163-170 (1960), who showed that for h >= 12, if we put g = (h+3)/2 for odd h and g = (h+4)/2 for even h, and if p^g divides |G|, then p^h divides |Aut(G)|. I have not found any more recent results than these. Derek Holt.