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

1,3

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

Links

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

Example

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, ...

Prog

(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

Crossrefs

Cf. A061350.

Sequence in context: A110633 A240232 A119250 * A127399 A240751 A212283

Adjacent sequences: A059770 A059771 A059772 * A059774 A059775 A059776

Keyword

nonn,nice

Author

Victor S. Miller, Feb 21 2001

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

Status

approved