A126103 Number of pointed groups of order n: that is, Sum_{G = group of order n} Number of orbits in G under the full automorphism group of G.

1, 2, 2, 5, 2, 7, 2, 17, 5, 7, 2, 23, 2, 7, 4, 67, 2, 23, 2, 25, 8, 7, 2, 99, 5, 7, 18, 20, 2, 25, 2, 342, 4, 7, 4, 89, 2, 7, 8, 99, 2, 40, 2, 20, 10, 7, 2, 476, 5, 23, 4, 25, 2, 100, 10, 87, 8, 7, 2, 115, 2, 7, 24, 2602, 4, 25, 2, 25, 4, 25, 2, 461, 2, 7, 13, 20, 4, 40, 2, 504, 79, 7, 2, 141, 4, 7, 4, 83, 2, 83, 4, 20, 8, 7, 4

Offset

1,2

Comments

Number of pairs (G, g in G) for G a group of order n, g an orbit representative for action of Aut(G) on G.

This has the same relation to A000001 (groups) as A000081 (pointed trees, also called rooted trees) does to trees (A000055).

Links

Klaus Brockhaus, Table of n, a(n) for n=1..191

Prog

(Magma) D:=SmallGroupDatabase();
for o in [1..95] do
t1:=0;
t2:=NumberOfSmallGroups(D, o);
for n in [1..t2] do
G:=SmallGroup(D, o, n);
H:=AutomorphismGroup(G);
gg:=[];
for g in G do Append(~gg, g);
end for;
PH:=[];
for h in Generators(H) do
ph:=[];
for i in [1..#gg] do
j:=Position(gg, gg[i]@h);
Append(~ph, j);
end for;
Append(~PH, ph);
end for;
pH:=sub<SymmetricGroup(#gg) | PH>;
t1:=t1 + #Orbits(pH);
end for;
print(t1);
end for;
(Magma) D:=SmallGroupDatabase(); [ &+[ #Orbits(sub<SymmetricGroup(o) | [ [ Position(gg, h(gg[i])): i in [1..o] ] where gg is [g: g in G] : h in Generators(AutomorphismGroup(G)) ] where G is SmallGroup(D, o, n) > ) : n in [1..NumberOfSmallGroups(D, o)] ] : o in [1..95] ]; /* Klaus Brockhaus, Mar 08 2007 */

Crossrefs

Cf. A000001 (groups). See A126102 for a different and somewhat inferior version.

Sequence in context: A256612 A305797 A029648 * A316895 A100030 A029603

Adjacent sequences: A126100 A126101 A126102 * A126104 A126105 A126106

Keyword

nonn

Author

Gabriele Nebe and N. J. A. Sloane, Mar 06 2007

Status

approved