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Maximum size of Aut(G) where G is a finite group of order n.
4

%I #22 Mar 02 2013 17:20:31

%S 1,1,2,6,4,6,6,168,48,20,10,24,12,42,8,20160,16,432,18,40,42,110,22,

%T 336,480,156,11232,84,28,120,30,9999360,20,272,24,864,36,342,156,672,

%U 40,252,42,220,192,506,46,40320,2016,12000,32,312,52,303264,110,1008

%N Maximum size of Aut(G) where G is a finite group of order n.

%C If n = 2^k then take G to be (Z/2Z)^k, the Abelian group with n=2^k elements and characteristic two. It is generated by any k linearly independent (non-identity) elements, so the automorphism group has size (n-1)(n-2)(n-4)...(n-2^(k-1)), which grows as n^log n. I think one can show that this is optimal for n=2^k and furthermore that this has the highest rate of growth for any infinite sequence of n's. - _Michael Kleber_, Feb 21 2001

%C Equals A061350(n) for n in A056867. - _Eric M. Schmidt_, Mar 02 2013

%H Eric M. Schmidt, <a href="/A059773/b059773.txt">Table of n, a(n) for n = 1..767</a>

%e The corresponding groups are 1, Z2, Z3, (Z2)^2, Z5, S3, Z7, (Z2)^3, (Z3)^2, D5, Z11, A4, Z13, D7, Z15, (Z2)^4, Z17, ...

%o (GAP) A059773 := function(n) local max, f, i; if IsPrimePowerInt(n) then f := PrimePowersInt(n); return Product([0..f[2]-1], k->n-f[1]^k); fi; max := 1; for i in [1..NumberSmallGroups(n)] do max := Maximum(max, Size(AutomorphismGroup(SmallGroup(n,i)))); od; return max; end; # _Eric M. Schmidt_, Mar 02 2013

%Y Cf. A061350.

%K nonn,nice

%O 1,3

%A _Victor S. Miller_, Feb 21 2001

%E More terms from Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 09 2001

%E a(18)-a(56) from _Stephen A. Silver_, Feb 26 2013