A059773 Maximum size of Aut(G) where G is a finite group of order n.

1, 1, 2, 6, 4, 6, 6, 168, 48, 20, 10, 24, 12, 42, 8, 20160, 16, 432, 18, 40, 42, 110, 22, 336, 480, 156, 11232, 84, 28, 120, 30, 9999360, 20, 272, 24, 864, 36, 342, 156, 672, 40, 252, 42, 220, 192, 506, 46, 40320, 2016, 12000, 32, 312, 52, 303264, 110, 1008

Offset: 1

Author

Victor S. Miller, Feb 21 2001

Keywords

• nonn
• nice

Comments

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

Equals A061350(n) for n in A056867. - Eric M. Schmidt, Mar 02 2013

Examples

			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, ...
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..767

Crossrefs

Cf. A061350.

Programs

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

Extensions

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

a(18)-a(56) from Stephen A. Silver, Feb 26 2013